Geometry - Math Workout for the GRE

Math Workout for the GRE, 3rd Edition (2013)

Chapter 7. Geometry

ANGLING FOR A BETTER GRADE?

Now that you’ve reached this chapter, you may be having a flashback to your freshman year in high school when you first came in contact with theorems, postulates, and definitions, all woven together to form the geometric proof. Well, relax. GRE geometry questions have little to do with deductive reasoning. You’re much more likely to be tested on the basic formulas involving area, perimeter, volume, and angle measurements. As you work through the GRE math, you’ll find that there is a basic battery of terms and formulas that you should know for the geometry questions that do come your way. Before we get to those, let’s look at some techniques.

· Plug In. If a problem tells you that a rectangle is x inches long and y inches wide, Plug In some real numbers to help the question take on a more tangible quality. (And if there is more than one variable, remember to Plug In a different number for each one.)

· Use Ballparking. If a diagram is drawn to scale, you can sometimes estimate the right answer and eliminate all the answer choices that don’t come close.

· Re-draw to Scale. If a diagram is not drawn to scale—and for problem solving questions you’ll know because you’ll see the words “Note: Figure not drawn to scale” right below the picture—re-draw it to make it look like it’s supposed to look like. Drawings like this are meant to confuse you by suggesting that the figure looks how it’s represented in the problem. Re-drawing the figure to scale helps you avoid falling into that trap.

Drawn to Scale:
Problem Solving

Problem Solving figures
are typically drawn to
scale. When they are not
drawn to scale, ETS adds
“Note: Figure not drawn to
scale” beneath the figure.

Drawn to Scale:
Quant Comp

Quant Comp figures are
often drawn to scale but
sometimes they aren’t to
scale. If they aren’t, ETS
does not add any sort of
warning like they do for
problem solving. Check
the information in the
problem carefully and be
suspicious of the figure.

BASIC HINTS FOR GEOMETRY QUESTIONS

Geometry is a special science all its own, but that doesn’t mean it marches to the beat of an entirely different drummer. Many of the techniques you’ve learned for other problems will work here as well.

So let’s start with our pal Euclid and his three primary building blocks of measured space: points, lines, and planes.

LINES AND ANGLES

Two points determine a line, and two intersecting lines form an angle measured in degrees. There are 360° in a complete circle, so halfway around the circle forms a straight angle, which measures 180°, and half of that is a right angle, which measures 90°.

Two lines that intersect in a right angle are perpendicular, and perpendicularity is denoted by the symbol “.” Two lines that lie in the same plane and never intersect are parallel, which is denoted by “||.” Take a look at two parallel lines below:

If one line intersects two parallel lines, it is called a transversal. It may look as though this transversal creates eight angles with the two lines, but there are actually only two types: big angles and small angles. All the big angles have the same degree measure, and all the small angles have the same degree measure. The sum of the degree measures of one big angle and one small angle is always 180°.

Trigger: Two parallel lines
cut by a transversal.

Response: Label all acute
(small) angles as equal,
and all obtuse (large)
angles as equal.

Notice that for the figure above, the acute (small) angles labeled 1, 3, 5, and 7 are all the same, because l1 is parallel with l2. We also know that the angles labeled 2, 4, 6, and 8 are all the same for the same reason. Whenever the GRE states that two lines are parallel, look to see if the question is actually testing this concept.

TRIANGLES

Three points determine a triangle, and all triangles have three sides and three angles. The sum of the measures of the angles inside a triangle is 180°. The sides and angles are related. Just remember that the longest side is always opposite the largest angle, and the shortest side is always opposite the smallest angle.

Types of Triangles

The properties of triangles start to get more interesting when some or all of the sides have the same length.

· Isosceles Triangles: If two sides of a triangle have the same length, the triangle is isosceles. The relationship between sides and angles still goes; if two sides of a triangle are the same length, then the angles opposite those sides have the same degree measure.

· Equilateral Triangles: Equilateral triangles have three equal sides and three equal angles. Because the sum of the angle measures is 180°, each angle in an equilateral triangle measures or 60°.

Right Triangles

Right triangles contain exactly one right angle and two acute angles. The perpendicular sides are called legs, and the longest side (which is opposite the right angle) is called the hypotenuse.

The Pythagorean Theorem

Whenever you know the length of two sides of a right triangle, you can find the length of the third side by using the Pythagorean theorem.

Trigger: Need to know
the side of a right triangle.

a2 + b2 = c2

Response: Write down
a2 + b2 = c2 and Plug In the
two sides you know.

Most of the time, one of the side lengths of a right triangle is irrational and in the form of a square root. Any set of three integers that works in the Pythagorean theorem is called a “Pythagorean triple,” and they’re very useful to know for the GRE because they come up often. The most common triple is 3 : 4 : 5 (because 32 + 42 = 52), but the other three worth memorizing are 5 : 12 : 13, 7 : 24 : 25, and 8 : 15 : 17.

All multiples of Pythagorean triplets also work in the Pythagorean theorem. If you multiply 3 : 4 : 5 by 2, you get 6 : 8 : 10, which also works.

Special Triangles

Two specific types of right triangles are called “special” right triangles because their angles and sides have measurements in a fixed ratio. The first, an isosceles right triangle, is also referred to as a 45°−45°−90° triangle because its angle measures are 45°, 45°, and 90°. The ratio of its side lengths is 1 : 1 : . The second is a 30°−60°−90° triangle, which has side lengths in a ratio of 1 : : 2.

ETS likes to use special triangles because they can confuse test takers into thinking they don’t have enough information to answer a question. The fact is, though, if you know the length of one side of a special triangle, you can use the ratios to find the lengths of the other two sides.

Question 18 of 20

In the figure shown above, if the height of triangle MPO is 4 inches, what is the perimeter, in inches, of triangle MPO ?

8 + 4 + 4

12 + 8

12 + 4 + 4

20 + 8

20 + 4 + 4

Here’s How to Crack It

The figure consists of two special triangles; consider the 45°−45°−90° triangle on the left. If PN = 4, then MN = 4 and MP = 4. The height PN also helps you find the lengths of the 30°−60°−90° triangle on the right. Because PNis the short side, the hypotenuse PO is twice as long, or 8 inches long. The other side, NO, measures 4. The perimeter of triangle MPO is therefore 4 + 4 + 8 + 4 so the answer is choice (C).

Area

The formula for the area of a triangle is A = bh, where b is the length of the base and h is the perpendicular distance from the vertex to the base (also known as the height, or the altitude). Because the legs of a right triangle are perpendicular, you can use the length of one leg as the base and the length of the other as the height.

Trigger: Triangle question
contains the word “area.”

Response: Write down
A = bh and Plug In what
you know. The height will
always be perpendicular
to the base.

In the diagram above, each of the triangles has the same base and a height of the same measure. Therefore, each triangle has the same area.

Triangle Quick Quiz

Question 1 of 6

If the degree measures of two angles of ∆ABC are 50° and 65°, what is the degree measure of the third angle?

15°

50°

65°

115°

165°

Question 2 of 6

If isosceles ∆DEF has sides of length 11.5 and 13.7, which of the following could be the perimeter of the triangle?

Indicate all such values.

2.1

12.0

25.2

36.7

50.4

Question 3 of 6

The base of a triangle is twice its height, which is 5 cm. What is the area, in square centimeters, of the triangle?

6.25

10

12.5

25

50

Question 4 of 6

What is the perimeter of triangle XYZ shown above?

Question 5 of 6

On his way home from shopping, Mike must travel due south for 5 miles, then due east for another 12 miles to reach his house. If Mike could travel in a straight line from the store to his house, how many fewer miles would he travel?

1

4

8

13

17

Question 6 of 6

If the lengths of all 3 sides of triangle RST are distinct, single-digit prime numbers, then which of the following could be the perimeter of triangle RST ?

Indicate all such values.

6

10

12

14

15

19

Explanations for Triangle Quick Quiz

1. The first two measures are 50° and 65°, so their total measure is 115°. The third angle must measure 180 − 115, or 65°. The answer is (C).

2. Since the triangle is isosceles, then third side must be either 11.5 or 13.7. If the third side is 11.5, then the perimeter is 36.7. If the third side is 13.7, then the perimeter is 38.9. The answer is (D).

3. The height is 5 cm so the base is 10 cm. The area is therefore (5)(10), or 25 cm2. The answer is (D).

4. Using the Pythagorean Theorem, you can find the third side. 252 − 152 = 400, and the square root of 400 is 20. The perimeter equals 15 + 20 + 25, or 60. You can also find the third side based on the 3-4-5 Pythagorean triple (just multiply the whole ratio by 5). The answer is 60.

5. Draw out the right triangle described in the question. By traveling due south and then due east, you get a right triangle with legs of length 5 and 12. Now you can solve for the length of the hypotenuse, or the distance from the store to the house. It equals 13 from the 5-12-13 Pythagorean triple. The question asks how many fewer miles Mike would travel if he could travel in a straight line. Mike travels a total of 5 + 12 = 17 miles as opposed to 13 miles if he could travel in a straight line. 17 − 13 = 4, so he would save 4 miles, which is choice (B).

6. The only single-digit prime numbers are 2, 3, 5, and 7; remember, though, that triangles must conform to the Third Side Rule, which states that the largest side of a triangle must be less than the sum of the other two sides. Only choice (E) is the sum of 3 sides that meet both requirements: 3, 5, and 7. Choices (B), (C), and (D) can also be reached by adding 3 distinct single digit primes—2, 3, and 5; 2, 3, and 7; and 2, 5, and 7, respectively—but all violate the Third Side Rule. If you selected choice (A) or choice (F), remember to only use distinct values for the sides. The answer is (E).

How Long Is the Third Side?

If you know the lengths of two sides of a triangle, you can use a simple formula to determine how long and how short the third side could possibly be.

If the lengths of two sides of a triangle are x and y, respectively, the length of the third side must be less than x + y and greater than |xy|.

Question 17 of 20

The townships of Addington and Bordenview are 65 miles apart, and Clearwater is 40 miles from Bordenview. If the three towns do not lie on a straight line, which of the following could be the distance from Addington to Clearwater?

15

25

35

105

125

Here’s How to Crack It

This is a “third-side” problem disguised as a word problem about three towns. Because the towns do not lie along a straight line, they form a triangle; one side is 65 miles long, the other is 40 miles long. Therefore, the third side (the length between towns A and C) must be greater than 65 − 40, or 25, and less than 65 + 40, or 105. (Remember, the distance has to be greater than 25, so it can’t be equal to 25, nor can it be equal to 105.) Therefore, the correct answer is (C).

QUADRILATERALS

A quadrilateral is any figure that has four sides, and the same types of quadrilaterals—parallelograms, rectangles, and squares—show up over and over again on the GRE. Regardless of their shape or size, however, one thing is true of all four-sided figures: They can be divided into two triangles. From this, we can determine a couple of things:

Degrees: Because every quadrilateral can be divided into 2 triangles, all quadrilaterals obey what we call the Rule of 360: There are 180 degrees in a triangle, so there are 2 × 180, or 360, degrees in every quadrilateral.

Area: The area of a triangle is bh, so the area of a parallelogram is 2×bh, or bh.

Properties, Area, and Perimeter

Quadrilaterals are often referred to as a “family” because they share lots of characteristics. For example, every rectangle is a parallelogram, so rectangles have every characteristic that a parallelogram has, and they also happen to have four right angles. Ditto for a square, which is just a rectangle that has four equal sides.

Trigger: Problem with
parallelogram, rectangle,
or square contains the
word “area.”

Response: Write down
the area formula and Plug
In information.

Trigger: Problem
mentions “perimeter.”

Response: Find the
length of each side and
add up all sides.

Here is a handy chart to help you keep track of all the various properties and the formulas for area and perimeter.

Quadrilateral Quick Quiz

Question 1 of 3

If the degree measures of two angles in a quadrilateral are 70° and 130° and the remaining two angles are equal to each other, what is the degree measure of one of these angles?

60°

80°

100°

160°

200°

Question 2 of 3

If the length of a rectangular garden is five times its width, and the perimeter of the garden is 36 feet, what is the garden’s width?

3

12

15

18

30

Question 3 of 3

If the base and height of a parallelogram are 10 cm and 15 cm, respectively, what is the area, in square cm, of the parallelogram?

25

50

75

150

It cannot be determined from the information given.

Explanations for Quadrilateral Quick Quiz

1. The sum of the given angles is 70° + 130°, or 200°, so the sum of the remaining angles must be 360° − 200°, or 160°. These angles are the same size, so they must each measure or 80°. The answer is (B).

2. If the width of the garden is w, then the length is five times that, or 5w. Plug these into the formula for perimeter (2l + 2w) and solve: 2(5w) + 2w = 36, so 12w = 36 and w = 3. The garden is 3 feet wide. The answer is (A).

3. The area of a parallelogram is base × height, so the parallelogram’s area is 10 × 15, or 150 cm2. The answer is (D).

CIRCLES

A circle represents all the points that are a fixed distance away from a certain point (called the center). The fixed distance from the center to the edge is the radius, and all radii are equal in length. When a radius is rotated 360° around the center, the circumference (the perimeter of the circle) is formed; any segment connecting two points on the circumference is called a chord. The diameter is the longest chord that can be drawn on a circle; it goes through the center and is twice as long as the radius.

Area and Circumference

Circles are wondrous things, because they gave us π. One day, a Greek mathematician with a lot of time on his hands began measuring circumferences (C) of circles and dividing those distances by the diameters (d), and he kept getting the same number: 3.141592 … He thought this was pretty cool, but also hard to remember, so he renamed it “p.” He was Greek, though, so he used the Greek letter p, which is π.

Trigger: Circle problem
contains the word
“circumference.”

Response: Write C = 2πr
or C = πd on your
scratch paper.

From this discovery we find that = π, and this can be rewritten as C = πd. This is the most common formula for finding the circumference of a circle. A diameter is twice as long as a radius (d = 2r), so you can also write the formula as C = 2πr. The formula for the area of a circle is A = πr2.

• Area of a circle = πr2

• Circumference of a circle = 2πr

Trigger: Circle problem
contains the word “area.”

Response: Write A = πr2
on your scratch paper.

Notice that the radius, r, is in both of those formulas? The radius is the most important part of a circle to know. Once you know the radius, you can easily find the diameter, circumference, or the area. So if you’re ever stuck on a circle question, find the radius.

One quick note about π. Although it is true that π ≈ 3.1415 (and so on and so on), you won’t have to use that too often on the GRE. Don’t worry about memorizing π beyond the hundredths digit: It’s 3.14. Even that is more precise than the GRE typically requires. Most answers are going to be in terms of π, which means that the GRE is much more likely to have 5π as an answer choice than it is to have 15.707. So don’t multiply out π unless you absolutely have to. Most of the time, each individual π will either cancel out or be in the answer choices.

Sectors and Arcs

An arc is a measurement around the circumference of a circle, and a sector is a partial measurement of the area of a circle. Both depend on the measure of the central angle, which has its vertex on the center of the circle.

We will figure out the length of an arc or the area of a sector by comparing it to the entire circle. If the central angle is 180°, then the arc must be half of the circumference, because 180° is half of the total 360° in the central angle of a circle. A sector made up of a 90° angle must be of the total area, because 90° is one fourth of 360°.

Question 14 of 20

In the diagram above, the clock shows that it is five minutes after 9 o’clock. If the radius of the clock is 9 inches, what is the area of the sector created by the hands of the clock starting at 9 and moving clockwise?

13.5π

27π

81π

54π

Here’s How to Crack It

There are 360° in a circle and 12 numbers on the face of a clock. Therefore, the measure of the central angle between each numeral on the clock (say, between the 12 and the 1) is , or 30°. There are four such central angles between the 9 and the 1, so the central angle is 4 × 30, or 120°. The radius of the circle is 9 inches, so the area of the whole clock is π(9)2, or 81π. To find the area of the sector, use the formula: 81π × = 27π. The correct answer is (C).

Circle Quick Quiz

Question 1 of 4

What is the radius of a circle that has a circumference of 6π inches?

2

3

4

6

12

Question 2 of 4

What is the area of a circle that has a circumference of 6π?

12π

36π

Question 3 of 4

A cherry pie with a radius of 8 inches is cut into six equal slices. What is the area, in square inches, of each slice?

Question 4 of 4

A cherry pie with a radius of 8 inches is cut into six equal pieces. What is the degree measure of the central angle of each piece of pie?

15°

30°

60°

90°

120°

Explanations for Circle Quick Quiz

1. Because C = πd, the diameter of a circle with a circumference of 6π inches is 6 inches and thus the radius is 3 inches. The answer is (B).

2. The diameter of the circle is 6, so the radius is half that, or 3. The area of the circle is π(3)2, or 9π square inches. The answer is (C).

3. The area of the pie is πr2, or 64π, so each slice has an area of square inches. This converts to . The answer is (C).

4. There are 360° in a circle, so the measure of each central angles is , or 60°. The answer is (C).

THE COORDINATE PLANE

You might find a smattering of questions about the x-y plane, otherwise known as the Cartesian or rectangular coordinate plane on the GRE. The primary skill you’ll need to possess in order to answer these questions is the ability to plot points by using the calibrations on the x- and y-axes. These two lines divide the space into four quadrants.

When plotting the point (x, y) on the coordinate plane

• start at the point (0, 0), which is also known as “the origin”

• move x units to the right (if x is positive) or left (if x is negative)

• move y units up (if y is positive) of down (if y is negative)

When you plot points on the Cartesian plane, you’ll most likely be asked to (1) find the distance between two of them, (2) find the slope of the line that connects the two points, or (3) find the equation of the line they define.

Distance in the Coordinate Plane

If you need to find the distance between two points in the coordinate plane, draw a right triangle. The hypotenuse of the triangle is the distance between the two points, and the legs are the differences in the x- and y-coordinates of the two points.

Question 11 of 20

If point A is at (2, 3) in the rectangular coordinate plane, and point B is at (−3, 15), what is the length of line segment ?

Here’s How to Crack It

Start by drawing a simple coordinate plane on your scratch paper. You don’t need to mark out each and every tick mark; this is just to get a rough idea of where our points are. Your drawing will probably look something like this:

Now let’s find the length of each leg. The bottom leg is the distance between our two x-coordinates. From −3 to 2 is a total of 5 units (or, |−3−2| = 5). Our height is the distance between our two y-coordinates. From 3 to 15 is 12 units (or, |3 − 15| = 12). So we have a triangle with sides of length 5 and 12. We can either use Pythagorean theorem to find the hypotenuse, or use the fact that we have a 5 : 12 : 13 triangle (one of the Pythagorean triplets), which means that the line is 13 units long.

Many two-dimensional distance problems are really Pythagorean theorem problems in disguise. The GRE will often hide this using questions in which people travel north/south and east/west.

Question 12 of 20

Mark and Kim are at the post office. To get home, Kim walks 3 miles north and 2 miles west and Mark walks 8 miles south and 2 miles east. What is the approximate straight-line distance between Kim’s house and Mark’s house?

10.1 miles

11.7 miles

12 miles

13.2 miles

15 miles

Here’s How to Crack It

Let’s draw a diagram, first. Nothing fancy, just enough to show us the location of the post office relative to the two houses. It will probably look something like this:

Now that we’ve got that drawn, let’s make our triangle. The trick here is that we can now completely ignore the post office. We just want to know the distance from Kim’s house to Mark’s house, so those will be the two points of our triangle, like so:

The total east/west distance is 4 miles, and the total north/south distance is 11 miles. Those are the legs of our triangle. We can plug those into the Pythagorean theorem, and we’ll get 42 + 112 = c2; 16 + 121 = c2; 137 = c2; ≃ 11.7, answer (B).

Slope Formula

To find the slope of a line, you need two distinct points on that line: (x1, y1) and (x2, y2). Notice the subscripts that designate the first point from the second point. It doesn’t matter which points you assign to which values as long as you’re consistent.

Slope of a line =

The most important part of slope, however, is to understand what it means. The slope is a measure of how much a line goes up or down on the y-axis (rise) as it goes over on the x-axis (run). In simpler language, the slope measures how slanted the line is. A positive slope means that the line rises up from left to right. A negative slope means that the line goes down from left to right. Notice that we always read the line from left to right, like reading a sentence. A slope of zero means that as we go over, the line never rises: It just remains a level, flat line. An undefined slope means that the line never runs over; it just goes up and up and up. (The slope is undefined because the difference in x-coordinates for any two points is 0, and we can’t divide by 0.)

Equation of a Line

Any line on the coordinate plane can be represented in the form y = mx + b, in which m is the slope of the line and b is the y-intercept. For example, the line y = 3x − 6 has a slope of 3 and intersects the y-axis at the point (0, −6).

Point-Slope Formula

If you know the slope m of a line and the coordinates of one of the points on that line (x0, y0), you can use the point-slope formula to determine the equation of the line: yy0 = m (xx0).

Coordinate Plane Quick Quiz

Question 1 of 5

What is the equation of a line that has a slope of −3 and that passes through the point (2, 1) ?

y = −3x + 1

y = −3x + 7

y = −3x + 11

y = 3x + 1

y = 3x + 5

Question 2 of 5

What is the slope of the line that passes through the points (−4, 3) and (6, −2)?

−2

2

4

Question 3 of 5

In the rectangular coordinate system, what is the distance between (−4, 3) and (6, −2)?

5

5

6

6

7

Question 4 of 5

What is the equation of the line that passes through the points (−4, 3) and (6, −2)?

y = −x + 1

y = −x + 2

y = x + 5

y = x + 7

y = x + 8

Question 5 of 5

At what point does the line y = 2x + 6 intersect the x-axis?

(0, −3)

(0, 6)

(−6, 0)

(−3, 0)

(3, 0)

Explanations for Coordinate Plane Quick Quiz

1. You have a point and the slope, so use the point-slope formula: (y − 1) = −3(x − 2). Simplified, this becomes y − 1 = −3x + 6. Add 1 to both sides, and your final answer is y = −3x + 7. The answer is (B).

2. Use the slope formula: . The answer is (B).

3. Draw a triangle on your scratch paper. The bottom of the triangle goes from −4 to 6, so the base is 10 units long. The height of the triangle goes from −2 to 3, so it is 5 units long. Plugging these lengths into the Pythagorean theorem, a2 + b2 = c2, gives us 52 + 102 = c2; 25 + 100 = c2; 125 = c2; c = = × = 5, answer (B).

4. You know the slope, and you can choose either point you were given. Use the point-slope formula: y − (−2) = − (x − 6). This simplifies to y = − x + 1. The answer is (A).

5. You are given the formula for the line, y = 2x + 6. When the line crosses the x-axis, the value of y, at that point, will be 0. Eliminate answer choices (A) and (B). Now plug 0 in for y in the equation for the line and solve for x. x = −3, so the point is (−3, 0) and the answer is (D).

VOLUME FORMULAS

Volume problems on the GRE are rare and will involve only a select few geometric solids. Formulas for those solids are below, and it’s best simply to memorize them, just in case a volume problem comes up.

Solid

Volume Formula

Rectangular prism


V = l × w × h

Right circular cylinder


V = πr2h

Space Diagonal

If you know the dimensions of a rectangular prism, you can determine the length of the greatest distance between any two points in that prism. This distance is called the space diagonal, which can be found using what looks like an extension of the Pythagorean theorem.

If the dimensions of a rectangular prism are a, b, and c, then the space diagonal d can be found using the formula d2 = a2 + b2 + c2.

SURFACE AREA

Predictably, the surface area of a three-dimensional figure is the sum of all of its surfaces. Almost all surface area questions on the GRE will deal with rectangular solids. Here is the formula for the surface area of a rectangular solid.

SA = 2lw + 2wh + 2lh

Geometry Drill

Question 1 of 14

Lines X, Y, and Z intersect to form a triangle. If line X is perpendicular to line Y and line X forms a 30-degree angle with line Z, which of the following is the degree measure of the angle formed by the intersection of lines Y and Z ?

20°

30°

45°

50°

60°

Question 2 of 14

Points (x, 5) and (−6, y), not shown in the figure above, are in Quadrants I and III, respectively. If xy ≠ 0, in which quadrant is point (x, y)?

IV

III

II

I

It cannot be determined from the information given.

Question 3 of 14

In the triangle above, what is the degree measure of the smallest angle?

10°

40°

45°

50°

60°

Question 4 of 14

The area of the shaded region is 3π. What is the radius of circle with center O ?

3

6

9

12

36

Question 5 of 14

In the figure above, if BD is perpendicular to AC and AC = 21, then AD =

12

13

20

21

25

Question 6 of 14

Which of the following expresses the area of a square region in terms of its perimeter p ?

Question 7 of 14

The figure above is a rectangular solid in which JK = 2, KL = 9, and LM = 9. What is the total surface area of the rectangular solid?

234

162

134

117

20

Question 8 of 14

The four corners of the face of a cube have coordinates (a, b), (a, d), (c, b) and (c, d). If a = 2 and c is an even number between 6 and 11, which of the following could be the surface area of the cube?

96

150

294

384

600

Question 9 of 14

In the figure above, if AC = CD, then r =

45 − s

90 − s

s

45 + s

60 + s

Question 10 of 14

Triangle ABC is an isosceles right triangle. If AB = AC, then the area of a square with a side length equal to twice the length of BC is how many times the area of triangle ABC ?

Question 11 of 14

Which of the following has the greatest value?

The area of a rectangle with length 11 and height 6.

The area of a right triangle with base length 10 and height 10.

The area of a square with diagonal 8.

The area of a circle with a radius of 5.

The area of an equilateral triangle with a side of 12.

Question 12 of 14

In the figure above, if q = 130 and p = 120, then r =

20°

60°

70°

80°

90°

Question 13 of 14

In the figure above, a square with side of length 3 is inscribed in a circle that is inscribed in a square. What is the area of the larger square?

Question 14 of 14

In the figure above, a circle with center O is inscribed in square WXYZ. If the circle has radius 3, then PZ =

6

3

6 +

3 +

3 + 3

EXPLANATIONS FOR GEOMETRY DRILL

1. E

Draw the figure. If X is perpendicular to Y, and X and Z form a 30° angle, you have a triangle with one 90° angle and one 30° angle. There are 180 degrees in a triangle, so the remaining angle must be 180° − 90° −30° = 60°.

2. A

If (x, 5) is in Quadrant I, then x has a positive value. If (−6, y) is in Quadrant III, then y is negative. A point with a positive x-coordinate and negative y-coordinate is in Quadrant IV. For example, if x = 3 and y = −5, (x, y) = (3, −5), which is in Quadrant IV.

3. B

Use the Rule of 180 for a line to find that the value of x is 180° − 140° = 40°. Next, apply the rule of 180 for a triangle to determine that 9y + 5y = 140°. Thus, y = 10, the angle represented by 5y is 50°, and the angle represented by 9y is 90°. That angle represented by x, which is 40°, is the smallest angle, making choice (B) the correct answer.

4. B

Because = , the shaded region is of the circle. Multiply 3π by 12 to find the area of the entire circle, 36π. Now put the values you know into the formula for the area of the circle: 36π = πr2. Solve for r to find the radius is 6. The answer is (B).

5. C

Since BD is perpendicular to AC, it cuts the large triangle into two right triangles. The small triangle on top, BCD, is the familiar 5-12-13 triangle, so BD = 12. Since AC = 21 and CB = 5, AB = 16. Now you have the two short sides of ABD, 12 and 16; use the Pythagorean Theorem or recognize the multiple of the 3-4-5 triangle to find that AD = 20.

6. D

Plug In letting p = 8. If the perimeter of the square is 8, then each side of the square equals 2. If a square has sides of length 2, then its area is 4. 4 is the target. Put p = 8 into the answer choices. Answer choice (D) is the only one that matches the target of 4: = 4.

7. A

To find surface area, add the areas of each face. The rectangle facing outward has sides of KL = 9 and LM = 9, so it has an area of 9 × 9 = 81. The face opposite this front face is identical, and the area of those two faces totals 162. Now look at the face on the left of the solid: It has sides of 2 and 9. The area of this face is 2 × 9 = 18, and again, the opposite face is the same, for a total area of 36. Finally, the bottom and top faces are each 2 × 9, for a total area of 36. Add up the areas of all faces to find the total surface area: 162 + 36 + 36 = 234, choice (A). Alternatively, the measurements can be plugged into the surface area formula, which also gives 234 as the total. In either case, choice (A) is correct.

8. D

First, draw the face of the cube in the coordinate plane.

Because a = 2 and c could be 8 or 10, one side of the square face of the cube could measure 6 or 8. Now take the possible areas for a single face, found by squaring the side, and multiply by 6 to account for the 6 equal faces of the cube: 62 × 6 = 36 × 6 = 216 and 82 × 6 = 64 × 6 = 384. Only 384 is an option, so the answer is (D).

9. A

Δ BCD + Δ ABD make the big Δ ACD. The problem tells you that AC = CD. These are the legs of the big Δ ACD. Because ∠ ACD is a right angle, and we know the two smaller angles of Δ ACD are equal (because the sides are equal), then each of the angles must be 45 degrees. There are variables in the answer choices, so Plug In for s and r. If s = 20, then r = 25. Only answer choice (A) works.

10. 16

First, draw triangle ABC, and Plug In sides; isosceles right triangle is another way of saying 45-45-90 triangle, so make the legs 10 and the hypotenuse, or BC, 10. Now the area of triangle ABC is 50. Since BC = 10, the square has a side of 20, and an area of 800. To find how many times the area of the triangle this represents, calculate 800 ÷ 50 = 16 times the area of triangle ABC.

11. D

Unfortunately you will have to calculate them all. (A) is length multiplied by width, which equals 66. (B) is bh, which is 50. Eliminate (B). The square in (C) has a side length of 8 because the diagonal of a square is the hypotenuse of a right isosceles triangle with a ratio of sides of x : x : x, so the area of the square is 64. Eliminate (C). (D) has an area of πr2 which in this case is 25π, which will be something slightly north of 75. Eliminate (A). For (E) the height of an equilateral triangle is the middle side of a 30 : 60 : 90 triangle which has a ratio of sides of x : x : 2x. In this case the area of the triangle, bh, is (12)(6). is less than two. 36 × 2 is still less than answer choice (D), so you can eliminate (E). The answer is (D).

12. C

If p = 120°, then by the rule of 180 for lines, the lower unnamed angle in the figure must be 180° − 120° = 60°. Because this unnamed angle is also the lower part of ∠q, then the lower part of ∠q is 60° and the upper part is 70° q = 130°. Because ∠r is opposite the upper portion of ∠q, r is also 70°.

13. 18

Start by drawing the diagonal of the smaller square. This diagonal is the hypotenuse of a right triangle created by the diagonal and two sides of the small square. Use the Pythagorean theorem to find the length of the hypotenuse: c = . Alternatively, if you recognized this as a 45°-45°-90° triangle, you would know the ratio of the sides is a : a : a, and you could find the hypotenuse using the ratio. This length is also the diameter of the circle, and you could find the hypotenuse using the ratio. This length is also the diameter of the circle, and the diameter of the circle is equal to the length of the sides of the big square. Use the formula for the area of a square, A = s2, to find the area of the larger square. A = (3)2 = 18.

14. E

Draw a diameter across the circle. The diameter is the same length as a side of the square. The radius of the circle is 3, the diameter is 6, and the side of the square is also 6. XZ forms a diagonal of the square, and it divides the square into two 45°-45°-90° right triangles. Because you know the side of the square is 6, you can use the 45°-45°-90° ratio (a : a : a) to find that the length of the diagonal is 6. Finally, PZ can be divided into two pieces, one from Oto P and one from O to Z. Line segment OP is equal to the radius of the circle, 3. Line segment OZ is equal to half the diagonal of the square, or half of 6. The length of PZ must be the two added together, or 3 + 3, choice (E).