1
Q
What is a Polygon?
A
- A two-dimensional, closed shape made of line segments
- Includes three-sided shapes (triangles), four-sided shapes (quadrilaterals) and other polygons with n sides (where n is five or more)
- Note that a circle is a closed shape but it is not a polygon because it does not contain line segments
2
Q
What is the perimeter?
A
Sum of the lengths of all sides
3
Q
What is a quadrilateral?
A
- Any figure with four sides
- Note that you can cut up any quadrilateral into two triangles by slicing them across the middle to connect opposite corners
4
Q
What are special types of quadrilaterals?
A
- Trapezoids
- Parallelograms
- Special Parallelograms (Rhombuses, Rectangles, Squares)
*Note a square is both a rhombus and a rectangle
5
Q
What is a parallelogram?
A
- Quadrilateral in which the opposite sides are parallel and equal
- Opposite angles are also equal and adjacent angles add up to 180
- Consists of a square and two identical right triangles
6
Q
What is the area of a parallelogram?
A
Area = (base)(height)
*With parallelograms, as with triangles and trapezoids, remember that the base and the height must be perpendicular to one another
7
Q
What is a rhombus?
A
- Quadrilateral in which all of the sides have the same length and in which the opposite angles are equal
- Every rhombus is a parallelogram, and a rhombus with right angles is a square
8
Q
What is the area of a rhombus?
A
Area = (Diagonal1 * Diagonal2) / 2, where the diagonals refer to the lengths of the lines drawn between opposite vertices in the rhombus
*NOTE: the diagonals of a rhombus are always perpendicular bisectors (meaning they cut each other in half at a 90 degree angle
9
Q
What is a rectangle?
A
- Have all the same properties of a parallelogram, plus one more – all four internal angles of a rectangle are right angles
- With rectangles, you refer to one pair of sides as the length and one pair of sides as the width
*Note: the diagonal of a rectangle cuts the rectangle into two equal right triangles, with all the properties you expect of right triangles
10
Q
What are the perimeter and area of a rectangle?
A
Perimeter = 2(width + length) Area = width * length
11
Q
What is a square?
A
- A rectangle in which all 4 sides are equal and all angles are 90 degrees
- Thus, knowing only one side of the square is enough to determine the perimeter and area of a square
*NOTE: Squares are like circles in that if you know one measure, you can find everything
12
Q
What are the perimeter and area of a square?
A
Perimeter = 4(Side) Area = Side^2
13
Q
What is a trapezoid?
A
- A quadrilateral with at least one pair of parallel sides
- The parallel sides are called bases and the other two sides are called the legs
*NOTE: a scalene trapezoid is a trapezoid with no sides of equal measure
14
Q
What is the area of a trapezoid?
A
Area = h * (Base1 + Base2)/2
- Height refers to the line perpendicular to the two bases, which are parallel
- Note that you often have to draw in the height
15
Q
What are the Interior Angles of a polygon?
A
- The angles that appear in the interior of a closed shape
- The sum of those angles depends only upon the number of sides in the closed shape:
- Sum of Interior Angles of Polygon = (n - 2) * 180, where n = the number of sides in the shape
16
Q
What is the sum of the interior angles of a triangle?
A
3 sides, 180 degrees
17
Q
What is the sum of the interior angles of a quadrilateral?
A
4 sides, 360 degrees
*NOTE: a quadrilateral can be cut into two triangles by a line connecting opposite corners
18
Q
What is the sum of the interior angles of a pentagon?
A
5 sides, 540 degrees
*NOTE: a pentagon can be cut into three triangles by two lines connecting opposite corners
19
Q
What is the sum of the interior angles of a hexagon?
A
6 sides, 720 degrees
*NOTE: a hexagon can be cut into four triangles by three lines connecting opposite corners
20
Q
What does Two-dimensional mean?
A
A shape containing a length and a width.
21
Q
What does Three-dimensional mean?
A
An object containing a length, a width, and a height.
22
Q
What is a rectangular solid?
A
A three-dimensional shape consisting of six faces, at least two of which are rectangles (the other four may be rectangles or squares, depending upon the shape’s dimensions)
23
Q
What is the Surface Area of a Rectangular Solid?
A
Surface area = the sum of the areas of all six faces
24
Q
What is the Volume of a Rectangular Solid?
A
Volume = length * width * height, where length, width, and height refer to the three dimensions of the rectangular solid.
25
What is a Cube?
A three-dimensional shape consisting of six identical faces, all of which are squares
26
What is the Surface Area of a Cube?
Surface area = the area of any one face multiplied by 6.
NOTE: only need to know length of one side to determine surface area
27
What is the Volume of a Cube?
Volume = s^3, where s refers to the length of any one side of the cube
NOTE: only need to know length of one side to determine volume
28
How many books, each with a volume of 100 in^3, can be packed into a crate with a volume of 5,000 in^3?
BEWARE OF THE GMAT VOLUME TRICK. It is tempting to answer 50 books. However, this is incorrect, because you do not know the exact dimension of each book!
One book might be 5*5*4 while another book might be 20*5*1 – both of which have a volume of 100 in^3. However, both have different rectangular shapes
Takeaway:
-When you are fitting 3-dimensional objects into other 3-dimensional objects, knowing the respective volumes is not enough – you must know the specific dimensions (length, width, height) of each object to determine whether the objects can fit without leaving gaps
29
Geometry Strategy Guide, Ch 1, Q 10. ABCD is a square picture frame. EFGH is a square inscribed within ABCD as space for a picture. The area of EFGH (for the picture) is equal to the area of the picture frame (the area of ABCD minus the area of EFGH). If AB = 6, what is the length of EF?
```
Area(EFGH) = Area(ABCD) – Area(EFGH)
2Area(EFGH) = Area(ABCD)
2Area(EFGH) = 6*6
Area(EFGH) = 18 = x^2
X = √18 = 3√2
```
30
What is a Triangle?
- A three-sided closed shape composed of straight lines
| - The interior angles add up to 180°
31
How do you determine the area of a triangle?
Area of triangle = (1/2)(b)(h)
- Base refers to the bottom side of the triangle (note that ANY side of the triangle could act as a base)
- Height always refers to a line drawn from the opposite vertex to the base
- The base and the height must be perpendicular (form 90 degree angle) to each other
*The height can be outside the triangle! (You just have to extend the base, but not for the purposes of calculating the base in the area formula)
32
What is a Vertex (singular) or Vertices (plural)?
An "angle" or place where two lines of a shape meet; for example, a triangle has three vertices and a rectangle has four vertices
33
What are the Legs of a Triangle?
The smaller sides of a triangle; usually used in describing a right triangle, in which there is one hypotenuse (the longest side) and two legs (the shorter sides)
34
What is the Hypotenuse of a Triangle?
The longest side of a right triangle. The hypotenuse is opposite the right angle.
35
What is a Right Triangle?
- A triangle that includes a 90°, or right, angle
- Given the lengths of any two of the sides of a right triangle, you can determine the length of the third side using the Pythagorean Theorem
- With 30-60-90 and 45-45-90 right triangles, you only need the length of one side to determine the lengths of the other sides
*Right triangles are essential for solving problems involving other polygons
36
What are the two key properties of the angles of a triangle?
(1) Sum of the three angles of a triangle equals 180
(2) Angles correspond to their opposite sides
- Largest angle is opposite the longest side, while the smallest angle is opposite the shortest side and vice versa
- Additionally, if two sides are equal, their opposite angles are also equal (isosceles triangles)
37
What is the Triangle Inequality Law?
If you are given two sides of a triangle, the length of the third side must lie between the difference and the sum of the two given sides
- Any side of a triangle must be less than the sum of the other two sides
- Any side of a triangle must be greater than the difference of the other two sides
38
How do you determine the third angle of a triangle if you know two angles of a triangle or can represent all three in terms of x?
The sum of the internal angles of a triangle must add up to 180 degrees. As a result, if you know two angles of a triangle, you can find the third angle
Or you might be given one angle and the other angles in terms of x, in which case you solve the equation for x
39
What is a right triangle?
- Any triangle in which one of the angles is a right angle (90 degrees)
- Every right triangle is composed of two legs and one hypotenuse (side opposite the right angle, or c)
* APPLY PYTHAGOREAN’S THEOREM TO DETERMINE LENGTH OF SIDES
40
What is the Pythagorean Theorem?
a^2 + b^2 = c^2
-The lengths of the three sides of a right triangle are related by the equation above, where a and b are the lengths of the sides touching the right angle, also known as legs, and c is the length of the side opposite the right angle, also known as the hypotenuse
* Only applies to right triangles
* You can always find the length of the third side of a right triangle if you know the lengths of the other two sides
41
What are the Pythagorean triplets?
```
A subset of right triangles in which all three sides have lengths that are integer values
a-b-c
3-4-5 or 6-8-10 or 9-12-15 or 12-16-20
5-12-13 or 10-24-26
7-24-25
8-15-17
9-40-41
```
Note: You can double, triple or otherwise apply a common multiplier to these lengths
42
What is an imposter triangle?
A non-right triangle with two sides equal to two parts of a Pythagorean triplet
Example: a triangle with one side equal to 3 and one side equal to 4 does NOT necessarily mean that the third side has a length of 5; this rule only applies to triangles that are KNOWN to be right triangles
43
What is an isosceles triangle?
- A triangle that has two equal angles and two equal sides (opposite the equal angles)
- An isosceles right triangle has one 90 degree angle (opposite to the hypotenuse) and two 45 degree angles (45-45-90 triangle)
44
What are the properties of a right isosceles triangle?
If the angles of a triangle are equal to 45, 45 and 90 degrees, then the lengths of the sides are proportional to x : x : x√2
Leg : Leg : Hypotenuse
X : X : X√2
*IMPORTANT: an isosceles triangle is equal to exactly one half of a square (i.e. two right isosceles triangles make up a square)
45
What is an equilateral triangle?
A triangle that has 3 angles (all 60 degrees) and 3 equal sides
46
What do we know about a triangle inscribed inside of a circle?
- If one length is equal to the diameter of the circle, then the triangle is a right triangle
- The angle opposite to the hypotenuse is equal to 90 degrees
- Otherwise the triangle is not a right triangle
47
What are the properties of a 30-60-90 triangle?
If the angles of a triangle are equal to 30, 60 and 90 degrees, then the lengths of the sides are proportional to x : x√3 : 2x
Short Leg : Long Leg : Hypotenuse
X : X√3 : 2X
48
Given that an equilateral triangle has a side length of 10, what is its height?
- The side of an equilateral triangle is the same as the hypotenuse of a 30-60-90 triangle
- The height of an equilateral triangle is the same as the long leg of a 30-60-90 triangle
- Thus, the long leg has a length of 5√3, which is the height of the equilateral triangle
49
What is the area of an equilateral triangle?
In addition to the standard area formula for triangles, equilateral triangles have a special formula for area:
Area = (√3 * S^2)/4, where S is the length of any side of the equilateral triangle.
50
What do you do if you see two equal sides in a triangle?
Set the opposite angles equal to each other
51
What do you do if you see two equal angles in a triangle?
Set the opposite sides equal
52
What is the Diagonal of a Square?
d = s√2, where s is a side of the square
*This can also be the face diagonal of a cube
53
What is the Main Diagonal of a Cube?
- The main diagonal of a cube is the one that cuts through the center of the cube
- The diagonal of a face of a cube is not the main diagonal
d = s√3, where s is an edge of the square
54
What is the measure of an edge of a cube with main diagonal of length √60?
```
√60 = s√3
s = √60 / √3 = √20
```
55
What is the diagonal of a rectangle?
To find the diagonal of a rectangle, you must know either the length and width or one dimension and the proportion of one to the other
Using Pythagorean’s theorem, c = √(a^2 + b^2)
56
What is the Main Diagonal of a Rectangular Solid?
- The main diagonal of a rectangular solid is the one that cuts through the center of the solid
- The diagonal of a face of the rectangular solid is not the main diagonal
- The main diagonal of a rectangular solid can be found by using the “Deluxe” Pythagorean Theorem: d^2 = x^2 + y^2 + z^2, where x, y, and z are the length, width, and height of the rectangular solid, and d is the main diagonal
57
What are Similar Triangles?
-Triangles in which the three corresponding angles are identical and the corresponding sides are in proportion
58
What are the properties of Similar Triangles?
(1) 2 Angles: It is only necessary to determine that two sets of angles are identical in order to conclude that two triangles are similar; the third set will be identical because all of the angles of a triangle always sum to 180 degrees
(2) If two similar triangles have corresponding side lengths (or heights or perimeters) in ratio a:b, then their areas will be in ratio a^2:b^2
59
What is the principle regarding Similar Figures?
For similar solids (e.g. quadrilaterals, pentagons, etc) with corresponding sides in ratio a:b, their volumes will be in ratio a^3:b^3
60
How do you determine the length of the third side of a right triangle if you know the other two sides?
(1) Can find the third side, either by using the full Pythagorean theorem or by recognizing a triplet
(2) Make that unknown side less than the sum of the other two sides but more than the difference
61
Geometry Strategy Guide, Ch 2, Q 3. Triangle A has a base of x and height of 2x. Triangle B is similar to Triangle A, and has a base of 2x. What is the ratio of the area of Triangle A to Triangle B?
1 to 4. You can use the shortcut or calculate the Area for each triangle in terms of x.
Ratio of Areas = a^2 + b^2 = 1^2 : 2^2 = 1 : 4
Area(A) = (1/2)(x)(2x) = x^2
Area(B) = (1/2)(2x)(4x) = 4x^2
Ratio of Areas = 1:4
62
Geometry Strategy Guide, Ch 2, Q 6. In Triangle ABC, AD = DB = DC. Given that angle DCB is 60 degrees and angle ACD is 20 degrees, what is the measure of angle x?
If AD = DB = DC, then the three triangular regions in this figure are all isosceles triangles. Therefore, you can fill in some of the missing angle measurements. Since you know that there are 180 degrees in the large triangle ABC, you can write the following equation:
x + x + 20 + 20 + 60 + 60 = 180
2x + 160 = 180
x = 10
63
Geometry Strategy Guide, Ch 2, Q 8. What is the area of an equilateral triangle whose sides measures 8cm long?
Area Equilateral Triangle = (√3 * S^2)/4 = 16√3
64
Geometry Strategy Guide, Ch 2, Q 10. The points of a six-pointed star consist of six identical equilateral triangles, with each side 4cm. What is the area of the entire star, including the center?
Look at the star as one big triangle and three smaller triangles.
Area Large Triangle = (1/2)(12)(6√3) = 36√3
Area Small Triangle = (1/2)(4)(2√3) = 4√3
Area Star = 36√3 + 3(4√3) = 48√3
65
What is a circle?
- Set of points that are equidistant from a central point
- A circle contains 360 degrees
```
Circumference = 2(pi)(r)
Area = (pi)(r^2)
```
66
What is a semi-circle?
Half of a circle; a semi-circle contains 180º, exactly half of the 360º in a circle.
67
What is the radius of a circle?
- Any line segment connecting the center and any point on the circle is a radius (usually labeled r)
- All radii in the same circle have the same length
68
What is the diameter of a circle?
-Line segment that passes through the center of a circle and connects two opposite points on the circle (usually labeled d)
d = 2 * r
69
What is a chord?
- Any line segment that connects two points on a circle
| - Any chord that passes through the center of the circle is called the diameter
70
What are the basic measurements of a circle and the underlying principle relating them?
(1) C = circumference = 2(pi)(r); distance around a circle
(2) d = diameter = 2*r; distance across a circle
(3) r = radius = d/2; half the distance across a circle
(4) A = Area = (pi)(r)^2; area of a circle
**If you know any of these values, you can determine the rest
71
What is the circumference of a circle?
Measure of the distance around a circle
C = 2(pi)(r)
OR
C = (d)(pi)
72
What is the area of a circle?
The space inside the circle (usually labeled A)
A = (pi)(r^2)
73
What is the volume of a sphere?
Volume = (4/3)(pi)(r^3)
74
What is a Revolution?
Revolution = Circumference
A point on the edge of a wheel travels one circumference in one revolution
75
What is the key rule to know about circles?
If you know one thing about a circle (e.g. area), then you can find out everything else about the circle by using the standard formulas (r, d, and C)
76
What is a sector?
Any fractional portion of a circle (i.e. a slice of the pie)
77
What is the area of a sector?
The proportion of the central angle to 360 is the same as the proportion of the area of the sector to the area of the circle
For example, if the central angle is 60, the proportion is 60/360 = 1/6. The sector, then, is 1/6 the area of the total area
78
What is the arc length?
The remaining portion of the circumference included in the sector
Arc Length = Central Angle/360 * 2(pi)(r)
79
What is the central angle?
- An angle whose vertex lies at the center point of a circle
- The degree measure between the two radii that form a sector
- The central angle tells you the fractional amount that the sector represents; but any of the three properties of a sector (e.g. central angle, arch length, and area) could be used
Central Angle / 360 = Sector Area / Circle Area = Arc Length / Circumference
80
How do you determine the perimeter of a sector?
Perimeter of sector = 2(r) + Arc Length
81
What is an Inscribed Angle?
- An angle that has its vertex on the circumference of the circle itself, and thus does not originate from the center of the circle
- The other two endpoints define the intercepted arc on the circle
82
What do we know when an inscribed angle and a central angle intercept the same arc?
The inscribed angle is exactly half of the central angle
83
What do we know about a circle with two inscribed angles with the same intercepted arc?
The two inscribed angles are congruent
84
What is an Inscribed Triangle?
- A triangle is said to be inscribed in a circle if all of the vertices of the triangle are points on the circle
* RULE: If one of the sides of an inscribed triangle is a diameter of the circle, then the triangle must be a right triangle; conversely, any right triangle inscribed in a circle must have the diameter of the circle as one of its sides (thereby splitting the circle in half)
85
If the hypotenuse of an inscribed triangle is the diameter of the circle, what kind of triangle is it?
Right triangle
86
What should you do when you encounter a sector? (e.g. if a central angle is 45 degrees and the radius if 5, what is the area?)
Figure out the fraction of the circle that the sector represents
```
Fraction = 45/360 = 1/8
Area = (1/8)(pi)(5^2) = (1/8)(pi)(25)
```
87
What is a cylinder?
Composed of two circles on either end of a rolled up rectangle
88
What is the surface area of a cylinder?
Surface Area = 2 Circles + Rectangle = 2(Pi * r^2) + 2(Pi)(r)(h).
89
The only information you need in order to find the surface area of a cylinder is:
(1) radius of cylinder
| (2) height of cylinder
90
What is the volume of a cylinder?
V = (pi)(r^2)(h), where V is the volume, r is the radius of the cylinder, h is the height of the cylinder, and Pi is a constant that equals approximately 3.14.
*NOTE: two cylinders can have the same volume but different shapes (and therefore each fits differently inside a larger object)
91
The only information you need in order to find the volume of a cylinder is:
(1) radius of cylinder
| (2) height of cylinder
92
What do we know about a circle inscribed inside a square or rectangle?
Circle inscribed in a square is tangent to all sides
Circle inscribed in a rectangle is tangent to two sides
93
Geometry Strategy Guide, Ch 3, Q 8. A Hydrogenator water gun has a cylindrical water tank, which is 80cm long. Using a hose, Jack fills his Hydrogenator with Pi cm^3 of his water tank every second. If it takes him 8 minutes to fill the tank with water, what is the diameter of the circular base of the gun’s water tank?
```
In 8 minutes, or 480 seconds, 480(Pi) cm^3 of water flows into the tank. Therefore, the volume of the tank is 480(Pi). You are given the height of 80, so you can solve for the radius:
V = (Pi)(r^2)(h)
480(Pi) = 30(Pi)(r^2)
r^2 = 16
r = 4
```
94
What is a line?
- Shortest distance between two points
| - As an angle, a line measures 180 degrees
95
What are parallel lines?
-Lines that lie in a plane and that never intersect no matter how far they are extended
96
What are perpendicular lines?
Lines that intersect at a 90 degree angle
97
What are the two major line-angle relationships?
(1) Angles formed by any intersecting lines
| (2) Angles formed by parallel lines cut by a transversal
98
What are the three important properties of intersecting lines?
(1) a + b + c + d = 360. Interior angles formed by intersecting lines form a circle, so the sum of these angles is 360 degrees
(2) a + b = 180, b + c = 180, c + d = 180 and d + a = 180. Interior angles that combine to form a line sum to 180 degrees (supplementary angles)
(3) a = c and b = d. Angles found opposite each other where these two lines intersect are equal (vertical angles)
*These rules apply to more than two lines that intersect at a point
99
Refer to picture. What is the exterior angle rule of a triangle?
The exterior angle of a triangle is equal to the sum of the two non-adjacent (or opposite) interior angles
a + b + c = 180 (sum of angles in a triangle)
b + x = 180 (supplementary angles)
Therefore, x = a + c
*FREQUENTLY TESTED ON THE GMAT IN DISGUISED FORM
100
What is a transversal?
A line that passes through two lines in the same plane at two distinct points
101
What are the interior angles of intersecting lines?
The interior angles of two intersecting lines form a circle of 360 degrees
102
What are vertically opposite angles?
Vertical means that they share the same vertex (or corner point)
Vertically opposite points are equal to each other
103
Refer to picture. How do you determine if two lines are parallel?
Two lines are parallel if one of the following is true:
(1) Corresponding angles (angles in matching corners) are equal (a = e)
(2) Alternate interior angles (on opposite sides of the transversal but inside the two lines) are equal (c = f)
(3) Alternate exterior angles (on opposite sides of the transversal but outside the two lines) are equal (b = g)
(4) Consecutive interior angles (on one side of the transversal but inside the two lines) add up to 180 (d + f = 180)
104
What do we know about parallel lines that are cut by a transversal?
- 8 angles are formed but there are only TWO different measures
- All acute angles (less than 90) are equal; and all obtuse angles (more than 90) are equal
- Any acute angle is supplementary to any obtuse angle
105
When two lines intersect, what do you know about the angles?
The opposite angles are equal
106
What do you know about the angles that form a straight line?
The angles sum to 180 degrees
107
What is a Coordinate Pair, or Ordered Pair?
(x,y)
- The values of a point on a number line
- The first number in the pair is the x-coordinate, which corresponds to the horizontal location of the point as measured by the x-axis
- The second number in the pair is the y-coordinate, which corresponds to the vertical location of the point as measured by the y-axis
108
What is the Origin of a Coordinate Plane?
The coordinate pair (0,0) represents the origin of a coordinate plane
109
What are the Types of Slopes?
A line can have one of four types of slope:
(1) positive
(2) negative
(3) zero
(4) undefined
110
What is the slope of a line?
m = Slope = Rise / Run = Change in y / Change in x
- This tells you how steep the line is and whether the line is rising or falling
- The numerator tells you how many units you want to move in the y direction (i.e. how far up or down you want to move)
- The denominator tells you how many units you want to move in the x direction (i.e. how far left or right you want to move)
111
What is the x-intercept?
- Point on the line at which y = 0
| - To find x-intercepts, plug in 0 for y
112
What is the y-intercept?
- Point on the line at which x = 0
| - To find y-intercepts, plug 0 in for x
113
What is the formula for a line?
y = mx + b
- m and b represent numbers (positive or negative)
- m represents the slope of the line
- b represents the y-intercept; it tells you where the line crosses the y-axis
114
What is a Linear Equation?
- An equation that represents a straight line, often expressed as Ax + By = C, where A, B and C are constants
- In all cases, a linear equation will never use terms such as x^2, sqrt(x), or xy
115
How would you represent x = 4 on a coordinate plane?
VERTICAL LINE
If you know that x = 4, then the point can be anywhere along a vertical line that crosses the x-axis at (4,0)
116
How would you represent y = -2 on a coordinate plane?
HORIZONTAL LINE
Every point two units below 0 fits this condition. These points form a horizontal line that crosses the y-axis at (0,-2). We do not know anything about the x coordinate
117
How would you represent x > 0 on a coordinate plane?
Every point to the right of the y axis has an x coordinate greater than 0. We do not know anything about y. Thus, the entire right-hand side of the y axis is shaded
118
How would you represent x > 0 AND y < 0 on a coordinate plane?
The bottom right quadrant is shaded (quadrant IV). So you know that the point lies somewhere in the bottom right quarter of the coordinate plane
119
How do you write an equation for a line with only two points?
Use the Point-Slope Formula: Y – Y1 = m(X – X1)
You can you use either point to replace the values of X1 and Y1
m = Change in Y / Change in X = (Ya – Yb) / (Xa – Xb)
Slope-Intercept Formula: y = mx + b
What are the steps for writing a linear equation with two given points?
(1) Find the slope of the line by calculating rise over run
(2) Plug the slope in for the m in the slop-intercept equation
(3) Solve for b, the y-intercept, by plugging the coordinates of one point into the equation
(4) Write the equation in the form y = mx + b
120
How do you determine the distance between two points on a coordinate plane?
(1) Draw a right triangle connecting the points
(2) Find the lengths of the two legs of the triangle by calculating the rise and the run
(3) Use Pythagorean’s theorem to calculate the length of the diagonal, which is the distance between the points
121
What is a quadrant?
- The four quarters of a coordinate plane are called quadrants
- Each quadrant responds to a different combination of signs of x and y
- The quadrants are always numbered starting on the top right and going counter-clockwise
Q1: contains positive x values, positive y values
Q2: contains negative x values, positive y values
Q3: contains negative x values, negative y values
Q4: contains positive x values, negative y values
122
What is the slope of a line that is perpendicular to another line?
Negative reciprocal. For example Line 2 is perpendicular to Line 1, and there slopes are:
```
m1 = 3
m2 = -1/3
```
123
What are the slopes of parallel lines?
Slopes of parallel lines are identical
124
For a square, if you know the value of any one of the following elements, you can calculate the rest
(1) side length
(2) diagonal length
(3) perimeter
(4) Area
125
For a circle, if you know the value of any one of the following elements, you can calculate the rest
(1) radius
(2) diameter
(3) circumference
(4) area
126
For a 30-60-90 or 45-45-90 triangle, if you know the value of any one of the following elements, you can calculate the rest
(1) one side length and its corresponding angle
(2) perimeter
(3) area
127
What are two major principles of GMAT geometry?
(1) DO NOT assume what you do not know for sure
| (2) Use every piece of information you do have to make further inferences
128
What is the process to solve geometry word problems?
(1) Draw or redraw figures
(2) Fill in the given information
(3) Identify the desired value
(4) Infer from the givens using equations and relationships for unknowns
(5) Solve the equations for the desired value
129
What are the bridges between inscribed shapes?
(1) Circle in Square: diameter = side length
(2) Triangle in Rectangle: length + width of rectangle = base + height of triangle
(3) Triangle in Trapezoid: height of trapezoid = height of triangle
(4) Triangle in Circle: radii of circle = sides of triangle
(5) Square in Circle = diameter of circle = diagonal of square
130
If one of the sides of an inscribed triangle is the diameter of the circle, then:
The triangle must be a right triangle, with the right angle lying opposite a semicircle
Conversely, any right triangle inscribed in a circle must have the diameter of the circle as one of its sides
131
What is a regular figure?
- Those that can change size but never change shape (e.g. squares, equilaterals, circles, spheres, cubes, 45-45-90 triangle, 30-60-90 triangles, and others)
- You only need ONE measurement in order to know EVERY measurement
- If you have two regular figures, and you know how they are related numerically (e.g. the side of an equilateral triangle has the same length as the diagonal of a square), then you can conclude that ANY measurement for EITHER figure will give you ANY measurement for either figure
132
Refer to picture. If lines m and n in the figure below are parallel, what is y?
Step 1: when geometry questions involve a figure, redraw it
Step 2: fill in the given information
Step 3: Identify the wanted element
Step 4: Infer from the givens using variables you already have
BAC + 3x = 180 --> BAC = 180 – 3x
BCA + 2x = 180 --> BCA = 180 – 2x
40 + (180 – 3x) + (180 – 2x) = 180
400 – 5x = 180
220 = 5x
x = 44
133
Refer to picture. Geometry Strategy Guide, Ch 6, Q 2. ABCD is a parallelogram. The ratio of DE to EC is 1:3. AE has a length of 3. If quadrilateral ABCE has an area of 21, what is the area of ABCD?
Break Quadrilateral ABCE into two pieces: a 3 by 2x rectangle and a right triangle with a base of x and a height of 3. Therefore, the area of Quadrilateral ABCE is given by the following:
(3*3x) + (3x)/2 = 9x + 1.5x = 10.5x
If ABCE has an area of 21, then 21 = 10.5x and x = 2. Quadrilateral is a parallelogram; thus, its area is equal to base*height or 4x*3.
A = 4(2)(3) = 24
134
Geometry Strategy Guide, Ch 6, Q 4. Triangle ABC is inscribed in a circle, such that AC is a diameter of the circle and BAC = 45 degrees. If the area of triangle ABC is 72 square units, how much larger is the area of the circle than the area of the triangle ABC?
Area of Circle is 72(Pi) – 72 square units larger than the area of triangle ABC
```
Triangle ABC is a 45-45-90 triangle, and the base and height are equal. Assign the variable x to represent both the base and height:
A = bh/2
72 = (1/2)(x^2)
144 = x^2
x = 12
```
The hypotenuse is equal to 12√2. Therefore, the radius is equal to 6√2 and the area of the circle is 72(Pi)
135
Refer to picture. Geometry Strategy Guide, Ch 6, Q 7. What is the area of triangle ABC?
(1) Side DC = 20
(2) Side AC = 8
Answer D. These triangles are similar. Any time two triangles each have a right angle and also share an additional right angle (or, in this case, the 90 degree span on either side of Point C, they will be similar). Since the two triangles are right triangles, if you had any two sides of triangle ABC, you could get the third. Because the two triangles are similar, you could use any two sides of triangle EDC (note that you already have DE = 16), as well as the ratio of one triangle’s size to the other, to get the third side of EDC as well as all three sides of ABC.
(1) SUFFICIENT. Side DC = 20. Use the 20 and the 16 to get, via the Pythagorean theorem, that side CE = 12. Then AC = 8. Thus, you have all three sides of EDC, plus the ratio of one triangle to the other (side AC = 8 matches up with side DE = 16; thus the smaller triangle is one-half the size of the larger)
(2) SUFFICIENT. Side AC = 8. Note that this gives you the same information as statement 1. If AC = 8, then CE = 12, and you can calculate all three sides of EDC
136
How might maximum/minimum area of polygon problems be presented on the GMAT?
(1) PS: “what is the maximum area of…?”
| (2) DS: “is the area of rectangle ABCD less than 30?”
137
How do you determine the maximum area of a quadrilateral?
- Of all quadrilaterals with a given perimeter, the square has the largest area (even in cases involving non-integer lengths)
- Of all quadrilaterals with a given area, the square has the minimum perimeter
*Both of these principles can be generalized for n sides: a regular polygon with all sides equal, will maximize area for a given perimeter and minimize perimeter for a given area
138
How do you determine the maximum area of a triangle or parallelogram?
- If you are given two sides of a triangle or parallelogram, you can maximize the area by placing those two sides perpendicular to each other
* Maximize area of rhombus by turning it into a square
139
What is a quadratic function and its different shapes?
- Any function in the form of f(x) = ax^2 + bx + c, where a, b and c are constants
- Can be plotted as a parabola in the coordinate plane
Positive value for a = curve opens upwards
Negative value for a = curve opens downwards
Large ǀaǀ = narrow curve
Small ǀaǀ = wide curve
140
How do you determine how many x-intercepts there are and what they are?
(1) solve for zero by factoring and solving the equation directly
(2) plug in points and draw the parabola
(3) use the quadratic formula
x = [-b ± √(b^2 – 4ac)] / 2a
The discriminant, b^2 – 4ac, tells you how many solutions exist
- Positive: two roots
- Zero: one root
- Negative: zero roots
141
What is a Parabola?
- The graph of a quadratic function
- A quadratic function is an equation containing exponents or roots
- For example, y = x^2 is a quadratic function
142
What is a Bisector?
A line or line segment that cuts a line segment exactly in half.
143
What is a Perpendicular Bisector?
- A line or line segment that cuts a line segment exactly in half and forms a 90 degree angle with that line
- Perpendicular bisector has the negative reciprocal slope of the line segment it bisects (ie the product of the two slops is -1; exceptions are vertical and horizontal lines)
144
If the coordinates of point A are (2,2) and the coordinates of point B are (0,-2), what is the equation of the perpendicular bisector of line segment AB?
(1) Find the slope of AB. m = (y1 – y2) / (x1 – x2) = (2 + 2) / (2 – 0) = 4/2 = 2
(2) Find the slope of the perpendicular bisector. m = negative reciprocal = -1/2
(3) Find the midpoint of AB. x coordinate will be the average of the x coordinates for points A and B and the y coordinate will be the average of the y coordinate for points A and B
A: (2,2)
Midpoint: (1,0)
B: (0, -2)
(4) Put the information together. To find the value of b, the y intercept, substitute the coordinates of midpoint for x and y
0 = -1/2(1) + b
b = 1/2
The perpendicular bisector of segment AB has the equation y = (-1/2)x + ½
145
What is the slope of the perpendicular bisector of line segment AB?
Slope = negative reciprocal = -1/m
146
How do you determine the midpoint between A and B?
- The average of the x coordinates for points A and B
- The average of the y coordinates for points A and B
[(x1 + x2)/2 , (y1+y2)/2]
147
At what point does the line represented by y = 4x – 10 intersect the line represented by 2x + 3y = 26?
```
2x + 3(4x – 10) = 26
2x + 12x – 30 = 26
14x = 56
X = 4
Y = 6
```
Takeaway:
- If two lines in a plane do not intersect, then the lines are parallel (ie there is no pair of numbers (x,y) that satisfies both equations at the same time
- Two linear equations might represent the same line in which case, infinitely many points (x,y) along the line satisfy the two equations
148
Geometry Strategy Guide, Ch 7, Q 5. The lengths of two shorter legs of a right triangle add up to 40 units. What is the maximum possible area of the triangle?
A right triangle is half of a rectangle. Constructing a right triangle with legs that add up to 40 is equivalent to constructing a rectangle with a perimeter of 80. Maximize the area of the rectangle. Each side should have length of 80/4 = 20. Thus, the maximum area of the rectangle is 400, and the maximum area of the triangle is one-half of that value, or 200 square units.
149
Refer to picture. Geometry Strategy Guide, Ch 7, Q 6. What is x in the diagram?
You can calculate the area of the solid triangle as (1/2)(12)(3) = 18
Next, use the side length of 7 as the base and write the equation for the area as (1/2)(7)(x)
```
18 = (1/2)(7)(x)
7x = 36
x = 36/7
```
150
Geometry Strategy Guide, Ch 7, Q 7. The line represented by the equation y = -2x + 6 is the perpendicular bisector of the line segment AB. If A has the coordinate (7,2) what are the coordinates for B?
The line segment AB must have a slope of 0.5 (negative reciprocal of -2)
```
Y = 0.5x + b
2 = 0.5(7) + b
b = -1.5
```
The line containing AB is y = 0.5x – 1.5. Find the point at which the perpendicular bisector intersects AB by setting the two equations equal to each other
-2x + 6 = 0.5 – 1.5
2.5x = 7.5
X = 3; Y = 0
The two lines intersect at (3,0), which is the midpoint of AB
Point B has coordinates (-1, -2)
151
Refer to picture. Geometry Strategy Guide, Ch 7, Q 10. The coordinates for point A are (-5,6) and the coordinates for point B are (-2,0). What are the coordinates for the point on line AB that is three times as far from A as from B, and that is between points A and B?
We have two similar triangles, such that the smaller one is equal to 1/4 of the larger one
Find the x coordinate: 4x = 3 --> x = 0.75 --> x-coordinate = -2.75
Find the y coordinate: 4x = 6 --> x = 1.50 --> y-coordinate = 1.50
(-2.75, 1.50) is the point that is three times as far from A as from B and that is between points A and B
152
Ch 10, Q 44. A rectangle has a perimeter of 10 and an area of 6. What are the length and width of the rectangle?
We know the perimeter of the rectangle is:
2W + 2L = 10
W + L = 5
We also know the area of the rectangle is:
(W)(L) = 6
```
Use substitution to solve for one side:
W = 5 – L
(5 – L)(L) = 6
5L – L^2 = 6
L^2 – 5L + 6 = 0
(L – 3)(L – 2) = 0
L = 3 OR L = 2
```
Either the length is 3 and the width is 2, or the length is 2 and the width is 3
153
Ch 10, Q 49. A square has an area of 49. What is the length of the diagonal line that connects two of the corners?
If the square has an area of 49, then (side)^2 = 49. That means that the length of the sides of the square is 7.
Now we can use the Pythagorean Theorem to find the length of the diagonal line, which is also the hypotenuse
7^2 + 7^2 = c^2
c = Sqrt(98) = Sqrt(2 * 7 * 7) = 7Sqrt(2)
154
What is the formula for the number of diagonals in a polygon?
of diagonals = n(n - 3) / 2
where n is the number of sides of the shape
155
Recognition: (i) two similar triangles have sides with a ratio of 1/2 (ii) two similar triangles have sides with a ratio 2
(i) Then the areas of the triangles will have the ratio of (1/2^2) = 1/4
(ii) Then the areas of the triangles will have the ratio 2^2 = 4
156
What are the rules for congruent triangles?
SSS: if two triangles have three congruent sides, then the triangles are congruent
SAS: if two triangles have two congruent sides and the angles formed by these sides are congruent, then the triangles are congruent
ASA: if two triangles have two congruent angles and the adjacent sides are also congruent, then the triangles are congruent
AAS: if two triangles have two congruent angles and one congruent side, then the triangles are congruent
157
How many triangles are formed by a regular hexagon?
6 equilateral triangles
158
What are the basic function transformations for plotting quadratic equations?
f(x) = ax^2 + bx + c,
(1) x^2 + 3 ---- graph moves up three units on y axis
(2) (x + 3)^2 ---- graph moves to the left three units on x axis
(3) Positive value for a = curve opens upwards
(4) Negative value for a = curve opens downwards
(5) Large ǀaǀ = narrow curve
(6) Small ǀaǀ = wide curve
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