Geometry Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 3

Question 41Question →

Angles $X$ and $Y$ are complementary angles. The measure of angle $X$ is $(4x - 10)^\circ$ and the measure of angle $Y$ is $(x + 20)^\circ$ . What is the measure of angle $X$ in degrees?

54

126

74

70

Show answer & explanation

Answer: 54

Answer

54

Complementary angles have a sum of 90 degrees. By adding the expressions for angle X and angle Y together, we establish the equation (4x - 10) + (x + 20) = 90. Combining like terms yields 5x + 10 = 90. Subtracting 10 from both sides leaves 5x = 80, and dividing by 5 gives x = 16. The question asks for the measure of angle X, which is given by the expression (4x - 10). Substituting 16 for x gives 4(16) - 10 = 64 - 10 = 54 degrees.

Step-by-Step Solution

1

Set up an equation based on the definition of complementary angles.

(4x - 10) + (x + 20) = 90

Complementary angles always sum to 90 degrees.

2

Combine like terms to simplify the equation.

5x + 10 = 90

Grouping the x terms (4x + x = 5x) and constant terms (-10 + 20 = 10) simplifies the expression.

3

Solve for x by isolating the variable.

5x = 80 \Rightarrow x = 16

Subtracting 10 from both sides and then dividing by 5 isolates x.

4

Substitute the value of x back into the original expression for angle X.

4(16) - 10 = 64 - 10 = 54

The question asks for the specific measure of angle X, not just the algebraic value of x.

Key Concept

Complementary angles sum to 90 degrees, and unknown angle measures can be found by setting up and solving a linear equation.

Question 42Question →

A circular swimming pool is surrounded by a concrete deck of uniform width. The circumference of the outside edge of the deck is $44\pi$ feet, and the circumference of the pool itself is $36\pi$ feet. What is the area of the concrete deck in square feet?

8\pi

16\pi

160\pi

640\pi

Show answer & explanation

Answer: $160\pi$

Answer

The area of the concrete deck is

160\pi

square feet.

To find the area of the concrete deck, we must calculate the area of the large circle (pool plus deck) and subtract the area of the small circle (pool). First, find the radius of the large circle by dividing its circumference by

2\pi

:

44\pi / 2\pi = 22

. Its area is

\pi(22)^2 = 484\pi

. Next, find the radius of the pool:

36\pi / 2\pi = 18

. Its area is

\pi(18)^2 = 324\pi

. Finally, subtract the inner area from the outer area:

484\pi - 324\pi = 160\pi

.

Step-by-Step Solution

1

Find the radius of the entire circular area (pool plus deck).

2\pi R = 44\pi

gives

R = 22

feet.

The area formula requires the radius, which is found by dividing the circumference by

2\pi

.

2

Find the radius of the pool itself.

2\pi r = 36\pi

gives

r = 18

feet.

We need the inner radius to calculate the area of the pool.

3

Calculate the total area of the pool and deck combined.

Area_{total} = \pi (22)^2 = 484\pi

square feet.

This is the area of the larger outer circle.

4

Calculate the area of the pool.

Area_{pool} = \pi (18)^2 = 324\pi

square feet.

This is the area of the inner circle.

5

Subtract the pool's area from the total area to find the area of the deck.

484\pi - 324\pi = 160\pi

square feet.

The deck is the ring-shaped region between the outer and inner circles.

Key Concept

Area of a Circular Ring (Annulus)

Question 43Question →

A rectangular window glass measures $24$ inches in length and $18$ inches in width. What is the perimeter of the window glass, in inches?

42

66

84

432

Show answer & explanation

Answer: 84

Answer

84

The correct answer is

84

because the perimeter is the total distance around the boundary of the rectangle. Adding all four sides (

24 + 24 + 18 + 18

) or using the standard formula

P = 2L + 2W

gives

2(24) + 2(18) = 48 + 36 = 84

inches.

Step-by-Step Solution

1

Identify the correct formula for the perimeter of a rectangle.

P = 2L + 2W

or

P = 2(L + W)

The perimeter measures the total continuous distance around the outside edge of a shape.

2

Substitute the given length (

L = 24

) and width (

W = 18

) into the formula.

P = 2(24) + 2(18)

Plugging the known values into the equation sets up the calculation.

3

Perform the multiplication and addition to find the total distance.

48 + 36 = 84

Evaluating the mathematical expression yields the final perimeter in inches.

Key Concept

Perimeter of Rectangles

Question 44Question →

An architect is designing a circular observation deck with a circumference of $100\pi$ feet. The floor of the deck will be covered entirely with weather-resistant tiles. If one box of tiles covers $50\pi$ square feet, how many boxes are required to completely cover the floor?

2

50

100

200

Show answer & explanation

Answer: 50

Answer

50

The radius of the deck is found by dividing the circumference by

2\pi

(

100\pi \div 2\pi = 50

feet). The total area is then calculated using the formula

A = \pi r^2

, resulting in

2500\pi

square feet. Finally, dividing the total area by the coverage per box (

50\pi

) gives the required number of boxes, which is

50

.

Step-by-Step Solution

1

Calculate the radius of the circular deck using the given circumference.

r = 50

feet

The formula for circumference is

C = 2\pi r

. Substituting

100\pi

for

C

gives

100\pi = 2\pi r

, which simplifies to

r = 50

.

2

Calculate the total area of the observation deck.

Area = 2500\pi

square feet

The formula for the area of a circle is

A = \pi r^2

. Substituting

r = 50

gives

A = \pi(50)^2 = 2500\pi

.

3

Determine the number of boxes of tiles needed.

50

boxes

Divide the total area by the coverage of one box:

2500\pi \div 50\pi = 50

.

Key Concept

Perimeter and Area of 2D Figures

Estimated Time:1m 0s

Question 45Question →

A straight fence is intersected by a diagonal property line. The two adjacent angles formed on the same side of the straight fence are supplementary. If the measure of the smaller angle is represented by $x^\circ$ and the measure of the larger angle is represented by $(x + 40)^\circ$ , what is the measure of the larger angle?

65^\circ

70^\circ

110^\circ

150^\circ

Show answer & explanation

Answer: $110^\circ$

Answer

The measure of the larger angle is

110^\circ

.

Because the angles are supplementary, their sum must be exactly

180^\circ

. By creating the equation

x + (x + 40) = 180

, we combine the variables to get

2x + 40 = 180

. Subtracting 40 from both sides leaves

2x = 140

, making

x = 70

. Since the larger angle is defined as

(x + 40)^\circ

, adding 70 and 40 gives the correct measure of

110^\circ

.

Step-by-Step Solution

1

Set up an equation based on the definition of supplementary angles.

x + (x + 40) = 180

Adjacent angles on a straight line are supplementary and must sum to 180 degrees.

2

Combine like terms and isolate the variable

x

.

2x + 40 = 180 \Rightarrow 2x = 140 \Rightarrow x = 70

To solve for

x

, first subtract 40 from both sides, then divide by 2.

3

Calculate the measure of the larger angle using the expression given in the problem.

(70) + 40 = 110

The question asks for the larger angle, which is

(x + 40)^\circ

, not just the value of

x

.

Key Concept

Supplementary Angles and Basic Algebra

Estimated Time:45s

Question 46Question →

Translate the following geometric relationships into algebraic equations and perform the first step of simplification. Pair each geometric description on the left with its corresponding simplified equation on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Match the complementary angles with

3x = 60

; the supplementary angles with

3x = 150

; the vertical angles with

x = 10

; and the adjacent angles on a straight line with

5x = 150

.

Each geometric term dictates a specific mathematical equality: 'complement' means the sum is

90^\circ

, 'supplement' and 'straight line' mean the sum is

180^\circ

, and 'vertical angles' means the measures are equal to each other. Applying these definitions and performing basic algebraic simplification yields the correct matches.

Step-by-Step Solution

1

Formulate and simplify the equation for the complementary angles.

The setup is

3x + 30 = 90

, which simplifies to

3x = 60

.

Complementary angles always add up to

90^\circ

.

2

Formulate and simplify the equation for the supplementary angles.

The setup is

3x + 30 = 180

, which simplifies to

3x = 150

.

Supplementary angles always add up to

180^\circ

.

3

Formulate and simplify the equation for the vertical angles.

The setup is

3x = 30

, which simplifies to

x = 10

.

Vertical angles (angles opposite each other at an intersection) are geometrically equal.

4

Formulate and simplify the equation for the adjacent angles on a straight line.

The setup is

3x + (2x + 30) = 180

. Combining the

x

terms gives

5x + 30 = 180

, and subtracting 30 gives

5x = 150

.

Angles that form a linear pair on a straight line must sum to

180^\circ

.

Key Concept

Translating Geometric Angle Relationships into Algebraic Equations

Question 47Question →

A surveyor is marking out a straight property boundary. Another reference line intersects this straight boundary, creating two adjacent angles. If the measure of the first angle is $(5x)^\circ$ and the measure of the second angle is $(4x + 81)^\circ$ , what is the value of $x$ ?

2

11

29

1

Show answer & explanation

Answer: 11

Answer

The value of x is 11.

Since the two adjacent angles form a straight line, they are supplementary and their measures must add up to 180 degrees. Setting up the equation

5x + 4x + 81 = 180

allows us to combine like terms to get

9x + 81 = 180

. Subtracting 81 from both sides gives

9x = 99

, and dividing by 9 results in

x = 11

.

Step-by-Step Solution

1

Identify the geometric relationship between the two angles.

The angles form a linear pair, meaning their sum is

180^\circ

.

Adjacent angles that form a straight line are always supplementary.

2

Set up an algebraic equation.

5x + (4x + 81) = 180

Substitute the given angle expressions into the supplementary angle relationship.

3

Combine the like terms on the left side of the equation.

9x + 81 = 180

Add the coefficients of the x terms (

5x + 4x = 9x

) while keeping the constant separate.

4

Isolate the variable term.

9x = 99

Subtract 81 from both sides of the equation to balance it.

5

Solve for x.

x = 11

Divide both sides by 9 to find the final value.

Key Concept

Supplementary Angles on a Straight Line

Question 48Question →

Angles $\angle P$ and $\angle Q$ are supplementary angles. The measure of $\angle P$ is given by the expression $(7x + 20)^\circ$ , and the measure of $\angle Q$ is given by $(3x - 10)^\circ$ . What is the measure of the angle that is complementary to $\angle Q$ ?

49^\circ

139^\circ

73^\circ

29^\circ

Show answer & explanation

Answer: $49^\circ$

Answer

The complementary angle measures

49^\circ

.

The sum of supplementary angles is

180^\circ

. Setting

(7x + 20) + (3x - 10) = 180

yields

10x + 10 = 180

, which simplifies to

10x = 170

, meaning

x = 17

. Substituting

x = 17

into the expression for

\angle Q

gives

3(17) - 10 = 41^\circ

. The complement of an angle sums to

90^\circ

, so the complementary angle is

90^\circ - 41^\circ = 49^\circ

.

Step-by-Step Solution

1

Set up an equation based on the definition of supplementary angles.

(7x + 20) + (3x - 10) = 180

Supplementary angles have a sum of

180^\circ

.

2

Combine like terms and solve for

x

.

10x + 10 = 180 \Rightarrow 10x = 170 \Rightarrow x = 17

This isolates the variable

x

to determine its numerical value.

3

Substitute the value of

x

into the expression for

\angle Q

.

\angle Q = 3(17) - 10 = 51 - 10 = 41^\circ

This finds the exact degree measure of

\angle Q

.

4

Calculate the complement of

\angle Q

.

90^\circ - 41^\circ = 49^\circ

Complementary angles have a sum of

90^\circ

.

Key Concept

Solving multi-step algebraic equations derived from the properties of supplementary and complementary angles.

Question 49Question →

Match each geometric angle relationship with the correct calculated value of $x$ .

Click a left item, then click its matching right item

Show answer & explanation

Answer

Match complementary to x = 19, vertical to x = 15, linear pair to x = 18, and alternate interior to x = 20.

Each geometric relationship dictates a distinct algebraic setup. Complementary angles sum to 90, vertical angles are set equal to each other, a linear pair sums to 180, and alternate interior angles are set equal to each other. Solving these linear equations gives the matching

x

values.

Step-by-Step Solution

1

Set up the equation for the complementary angles.

5x - 5 = 90

Complementary angles sum to

90^\circ

.

2

Solve for

x

in the complementary equation.

x = 19

Add 5 to both sides, then divide by 5.

3

Set up the equation for the vertical angles.

5x - 30 = 2x + 15

Vertical angles are congruent (equal in measure).

4

Solve for

x

in the vertical angles equation.

x = 15

Subtract 2x from both sides and add 30 to both sides, yielding 3x = 45.

5

Set up the equation for the linear pair.

8x + 36 = 180

Angles forming a linear pair are supplementary and sum to

180^\circ

.

6

Solve for

x

in the linear pair equation.

x = 18

Subtract 36 from both sides, then divide by 8.

7

Set up the equation for the alternate interior angles.

3x + 45 = 6x - 15

Alternate interior angles on parallel lines are congruent.

8

Solve for

x

in the alternate interior equation.

x = 20

Subtract 3x from both sides and add 15 to both sides, yielding 3x = 60.

Key Concept

Translating geometric angle relationships into algebraic equations to solve for unknown variables.

Question 50Question →

A straight road intersects two parallel train tracks. One of the acute angles formed at an intersection measures $72^\circ$ . What is the measure of any of the obtuse angles formed by these parallel tracks and the intersecting road, in degrees?

Show answer & explanation

Answer: 108

Answer

108

When a transversal intersects parallel lines, the sum of any acute angle and any obtuse angle formed is always

180^\circ

. Subtracting

72^\circ

from

180^\circ

gives the correct obtuse angle of

108^\circ

.

Step-by-Step Solution

1

Identify the relationship between the angles.

The acute and obtuse angles are supplementary.

When a transversal intersects parallel lines, consecutive interior angles (or any adjacent pair of acute and obtuse angles) add up to

180^\circ

.

2

Calculate the measure of the obtuse angle.

180^\circ - 72^\circ = 108^\circ

Subtracting the given acute angle from

180^\circ

yields the supplementary obtuse angle.

Key Concept

Parallel Lines and Transversals

Estimated Time:30s

Question 51Question →

A metal worker is joining two pieces of steel such that the two adjacent angles formed make a straight line. The measure of the first angle is $(2x + 10)^\circ$ and the measure of the second angle is $(3x)^\circ$ . What is the measure of the first angle?

78^\circ

42^\circ

58^\circ

34^\circ

Show answer & explanation

Answer: $78^\circ$

Answer

The measure of the first angle is

78^\circ

.

Angles that form a straight line are supplementary and add up to 180 degrees. Setting up the equation (2x + 10) + 3x = 180 yields 5x + 10 = 180. Solving for x gives x = 34. Substituting this value into the expression for the first angle gives 2(34) + 10 = 78 degrees.

Step-by-Step Solution

1

Set up an equation based on the properties of a straight line.

(2x + 10) + 3x = 180

Adjacent angles that form a straight line are supplementary, meaning their sum is exactly 180 degrees.

2

Combine like terms.

5x + 10 = 180

Adding the variable terms (2x and 3x) together simplifies the equation.

3

Solve for x.

x = 34

Subtract 10 from both sides to get 5x = 170, then divide by 5 to isolate x.

4

Substitute the value of x back into the expression for the first angle.

2(34) + 10 = 68 + 10 = 78

The question asks for the measure of the first angle, not just the value of x.

Key Concept

Supplementary Angles and Linear Equations

Question 52Question →

In quadrilateral $ABCD$ , $\angle B$ and $\angle D$ are right angles. The lengths of three sides are given as $AB = 7$ , $BC = 24$ , and $CD = 15$ . What is the length of side $AD$ ?

8

10

20

46

Show answer & explanation

Answer: 20

Answer

The length of side

AD

is 20.

Drawing the diagonal

AC

creates two right triangles that share a hypotenuse. The first triangle (

ABC

) has legs of

7

and

24

, giving a hypotenuse of

25

(

7^2 + 24^2 = 25^2

). In the second triangle (

ADC

), this

25

serves as the hypotenuse while

CD

is a leg of

15

. Solving for the remaining leg gives

20

(

25^2 - 15^2 = 400

, and

\sqrt{400} = 20

).

Step-by-Step Solution

1

Visualize or draw diagonal

AC

to split the quadrilateral into two right triangles.

Right

\triangle ABC

and right

\triangle ADC

are formed, sharing hypotenuse

AC

.

Since

\angle B

and

\angle D

are right angles, the segment connecting

A

and

C

must be the hypotenuse for both triangles.

2

Apply the Pythagorean theorem to

\triangle ABC

to find the length of

AC

.

AC^2 = 7^2 + 24^2 = 49 + 576 = 625

, which means

AC = 25

.

The lengths of both legs (

AB

and

BC

) are known, allowing calculation of the hypotenuse.

3

Apply the Pythagorean theorem to

\triangle ADC

using the known hypotenuse.

AD^2 + CD^2 = AC^2

, which becomes

AD^2 + 15^2 = 25^2

.

The hypotenuse

AC

is shared, and leg

CD

is given as 15, leaving

AD

as the only unknown variable.

4

Solve the equation for the missing leg

AD

.

AD^2 + 225 = 625 \Rightarrow AD^2 = 400 \Rightarrow AD = 20

.

Isolating

AD^2

and taking the square root provides the final length of the side.

Key Concept

Applying the Pythagorean theorem sequentially across adjacent right triangles that share a common hypotenuse.

Estimated Time:2m 0s

Question 53Question →

A drafter is reviewing a set of architectural plans. Connect each described geometric condition with its correct mathematical term.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Two angles summing to

180^\circ

match with Supplementary angles; two angles summing to

90^\circ

match with Complementary angles; opposite equal angles match with Vertical angles; lines forming a right angle match with Perpendicular lines.

Each definition precisely matches the fundamental geometric axioms for angles and lines. Recognizing these definitions is essential for solving more complex geometry problems.

Step-by-Step Solution

1

Identify the geometric term for two angles whose sum is exactly

180^\circ

.

Supplementary angles

By geometric definition, a pair of angles that add up to

180^\circ

(forming a straight line if adjacent) are supplementary.

2

Identify the geometric term for two angles whose sum is exactly

90^\circ

.

Complementary angles

By geometric definition, a pair of angles that add up to

90^\circ

(forming a right angle if adjacent) are complementary.

3

Identify the term for the opposite angles created when two lines intersect.

Vertical angles

Intersecting lines create two pairs of non-adjacent, opposite angles known as vertical angles, which are congruent.

4

Identify the term for straight lines that intersect to form a right angle.

Perpendicular lines

When lines intersect at exactly

90^\circ

, they are defined mathematically as perpendicular.

Key Concept

Basic terminology for angle relationships and line intersections

Estimated Time:45s

Question 54Question →

An isosceles triangle has a perimeter of $36$ centimeters. The base of the triangle measures $10$ centimeters. What is the area of the triangle in square centimeters?

Show answer & explanation

Answer: 60

Answer

60

To find the area, you must first know the perpendicular height. By splitting the isosceles triangle in half via its altitude, a right triangle is formed with a hypotenuse of

13

(the side length) and a base of

5

(half of the full base). The Pythagorean theorem (

5^2 + h^2 = 13^2

) reveals the height is

12

. The full area is then calculated as

\frac{1}{2} \times \text{base} \times \text{height}

, which is

\frac{1}{2} \times 10 \times 12 = 60

.

Step-by-Step Solution

1

Calculate the length of the two equal sides.

The two equal sides each measure

13

centimeters.

The perimeter is

36

cm and the base is

10

cm. The remaining

26

cm must be divided equally between the two identical legs:

26 \div 2 = 13

cm.

2

Determine the base of the right triangle formed by the altitude.

The base of the right triangle is

5

centimeters.

The altitude of an isosceles triangle bisects the base, splitting the

10

-centimeter base into two equal

5

-centimeter segments.

3

Calculate the height using the Pythagorean theorem.

The height is

12

centimeters.

In the right triangle formed, the hypotenuse is

13

and one leg is

5

. Let

h

be the height.

5^2 + h^2 = 13^2 \Rightarrow 25 + h^2 = 169 \Rightarrow h^2 = 144 \Rightarrow h = 12

.

4

Calculate the total area of the triangle.

The area is

60

square centimeters.

The area formula is

\frac{1}{2} \times \text{base} \times \text{height}

. Thus,

\frac{1}{2} \times 10 \times 12 = 60

.

Key Concept

Using the Pythagorean theorem to find missing dimensions in isosceles triangles to calculate area.

Question 55Question →

A circular target has a diameter of $10$ inches. What is the area of the target in square inches?

25\pi

100\pi

10\pi

5\pi

Show answer & explanation

Answer: $25\pi$

Answer

The correct answer is

25\pi

square inches.

The correct area is found by first dividing the diameter of

10

inches by

2

to get a radius of

5

inches. Squaring the radius gives

25

, which is then multiplied by

\pi

to yield

25\pi

square inches.

Step-by-Step Solution

1

Find the radius of the circular target.

r = \frac{10}{2} = 5

inches.

The area formula requires the radius, which is exactly half of the given diameter.

2

Apply the formula for the area of a circle.

A = \pi r^2 = \pi (5)^2 = 25\pi

.

Substitute the calculated radius into the standard area formula to find the final square footage.

Key Concept

Area of a Circle

Question 56Question →

A guy wire is attached to the top of a $12$ -foot vertical pole. The wire is anchored to the ground $5$ feet from the base of the pole. What is the length of the guy wire?

13

feet

\sqrt{119}

feet

17

feet

169

feet

Show answer & explanation

Answer: $13$ feet

Answer

The guy wire is

13

feet long.

The length of the guy wire represents the hypotenuse of a right triangle formed by the pole and the ground. According to the Pythagorean theorem (

a^2 + b^2 = c^2

), the square of the hypotenuse equals the sum of the squares of the two legs. Substituting the given values:

12^2 + 5^2 = 144 + 25 = 169

. Taking the square root of

169

gives

13

feet.

Step-by-Step Solution

1

Identify the known values representing the legs of the right triangle.

Leg

a = 5

feet, Leg

b = 12

feet.

The pole and the ground meet at a right angle, so they form the legs of the triangle.

2

Apply the Pythagorean theorem (

a^2 + b^2 = c^2

).

5^2 + 12^2 = c^2

The theorem connects the lengths of the legs to the hypotenuse (the guy wire).

3

Square the lengths of the legs and sum them.

25 + 144 = 169

This calculates the value of the hypotenuse squared (

c^2

).

4

Take the square root of the sum.

c = \sqrt{169} = 13

This isolates

c

and provides the actual length of the hypotenuse.

Key Concept

The Pythagorean theorem is used to find the hypotenuse of a right triangle when both legs are known.

Question 57Question →

A vertical flagpole that is originally $32$ feet tall breaks at a certain height due to strong winds. The top part of the pole folds over without completely detaching, and the tip touches the level ground exactly $16$ feet away from the base of the pole. At what height from the ground did the flagpole break?

8 feet

12 feet

16 feet

20 feet

Show answer & explanation

Answer: 12 feet

Answer

12 feet

The scenario creates a right triangle where the vertical leg is the standing part of the pole, the horizontal leg is the ground, and the hypotenuse is the broken top part. Letting the break height be

x

, the hypotenuse becomes

(32 - x)

. Using the Pythagorean theorem:

x^2 + 16^2 = (32 - x)^2

. Expanding the right side gives

x^2 + 256 = 1024 - 64x + x^2

. Subtracting

x^2

from both sides simplifies the equation to

256 = 1024 - 64x

. Adding

64x

and subtracting

256

gives

64x = 768

. Dividing by

64

yields

x = 12

. Therefore, the pole broke

12

feet above the ground.

Step-by-Step Solution

1

Define the variables for the sides of the right triangle formed by the broken flagpole.

Let the height of the break (standing part) be

x

. The length of the fallen part (hypotenuse) is the remainder of the pole, which is

(32 - x)

. The base distance along the ground is

16

feet.

Establishing these algebraic relationships is necessary to apply the Pythagorean theorem to the scenario.

2

Set up the Pythagorean theorem equation (

a^2 + b^2 = c^2

) using the defined variables.

x^2 + 16^2 = (32 - x)^2

The standing pole and the ground form a right angle, making the standing pole and ground distance the legs, and the fallen part the hypotenuse.

3

Expand both sides of the algebraic equation.

x^2 + 256 = 1024 - 64x + x^2

Expanding the squared binomial

(32 - x)^2

correctly is required to isolate the variable

x

.

4

Solve for

x

by combining like terms.

Subtract

x^2

from both sides to get

256 = 1024 - 64x

. Rearrange to

64x = 768

, and divide by

64

to find

x = 12

.

Isolating

x

provides the exact height from the ground where the flagpole snapped.

Key Concept

Applying the Pythagorean Theorem with algebraic expressions

Question 58Question →

Match each fundamental geometric term related to a circle with its correct definition.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Radius matches with a line segment from the center to the edge; Diameter matches with a line segment passing through the center; Circumference matches with the total distance around the circle; Chord matches with a line segment connecting any two points on the circle.

Each geometric term possesses a precise mathematical definition that distinguishes its function and location within a circle. Correctly identifying these fundamental components is an essential prerequisite for performing geometric calculations such as finding area or circumference.

Step-by-Step Solution

1

Identify the term that represents the perimeter or boundary distance of the circle.

Circumference matches with 'The total distance around the outer boundary of a circle.'

Circumference is the only term listed that measures a 1D distance around a shape rather than a straight interior line segment.

2

Differentiate between the line segments based on their relationship to the circle's center point.

Radius connects the center to the edge. Diameter passes all the way through the center from edge to edge.

This establishes the mathematical relationship where the diameter is exactly twice the length of the radius.

3

Determine the general term for a line segment connecting any two points on the circle's edge.

Chord matches with 'A line segment whose endpoints lie on the circle, but which does not necessarily pass through the center.'

A diameter is a special type of chord, but 'chord' is the broader category that does not require passing through the center.

Key Concept

Fundamental vocabulary and structural properties of circles.

Question 59Question →

A landscaper is placing a decorative border around a circular flower bed. The radius of the flower bed is $12$ feet. What is the total length of the border needed in terms of $\pi$ ?

24\pi

feet

12\pi

feet

144\pi

feet

576\pi

feet

Show answer & explanation

Answer: $24\pi$ feet

Answer

The total length of the border is

24\pi

feet, which is found by calculating the circumference of the circle.

To find the total length of the border, you must calculate the circumference of the circular flower bed. The formula for circumference is

C = 2\pi r

. Substituting the given radius of

12

feet into the formula gives

C = 2 \cdot \pi \cdot 12 = 24\pi

feet.

Step-by-Step Solution

1

Identify the mathematical concept required to find the length of a border around a circle.

The length of the border corresponds to the circumference of the circle.

Perimeter (circumference) measures the distance around the outside of a 2D shape.

2

Select the correct formula for the circumference of a circle.

Formula:

C = 2\pi r

The problem provides the radius (

r

), so this is the most direct formula to use.

3

Substitute the given radius value into the formula and solve.

C = 2 \cdot \pi \cdot 12 = 24\pi

feet.

Replacing

r

with

12

computes the exact circumference in terms of

\pi

.

Key Concept

Circumference of a circle

Question 60Question →

A hiker walks $8$ miles due north, then turns and walks $15$ miles due east. What is the straight-line distance, in miles, from the hiker's starting point to their current location?

Show answer & explanation

Answer: 17

Answer

17

Because the hiker's path consists of two perpendicular segments (north and east), they form the legs of a right triangle measuring

8

miles and

15

miles. The straight-line distance from the start to the end is the hypotenuse of this triangle. Applying the Pythagorean theorem (

a^2 + b^2 = c^2

), we get

8^2 + 15^2 = c^2

, which simplifies to

64 + 225 = 289

. The square root of

289

is

17

. Therefore, the straight-line distance is

17

miles.

Step-by-Step Solution

1

Visualize the geometrical shape formed by the hiker's path.

A right triangle is formed with legs of

8

miles and

15

miles.

The directions 'due north' and 'due east' are perpendicular, creating a

90^\circ

angle.

2

Set up the Pythagorean theorem equation.

8^2 + 15^2 = c^2

The straight-line distance corresponds to the hypotenuse (

c

) of the right triangle.

3

Square the values of the legs and add them together.

64 + 225 = 289

To evaluate the left side of the equation.

4

Calculate the square root of the sum.

c = \sqrt{289} = 17

To solve for

c

and find the final straight-line distance.

Key Concept

Calculating distance using the Pythagorean theorem

Geometry

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