Geometry Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 5

Match each 3D solid to its corresponding volume formula.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Rectangular Prism matches with

V = l \times w \times h

; Cube matches with

V = s^3

; Cylinder matches with

V = \pi r^2 h

; Sphere matches with

V = \frac{4}{3} \pi r^3

.

Each 3D solid has a distinct volume formula based on its geometry. Matching them correctly demonstrates fundamental knowledge required before solving applied numerical problems.

Step-by-Step Solution

1

Identify the base shape and defining dimensions for each 3D solid.

Rectangular prisms use length, width, and height. Cubes use a single side length. Cylinders use radius and height. Spheres use only radius.

Volume formulas are derived from the foundational properties and base shapes of the solids.

2

Connect the identified dimensions to the standard algebraic formulas.

All solids are successfully matched to their specific volume equations.

Accurate calculation of volume requires applying the correct formula for the specific geometric figure.

Key Concept

Volume Formulas of Basic 3D Solids

Estimated Time:45s

Question 82Question →

Calculate the specified length for the given shape.

Fill in the blanks below

A student cuts a piece of cardboard into a right triangle. The horizontal base measures

15

cm and the vertical height measures

20

cm. The length of the slanted side (hypotenuse) is cm.

Show answer & explanation

Answer

25

Applying the Pythagorean theorem (

a^2 + b^2 = c^2

), we substitute the legs

a = 15

and

b = 20

. Squaring these gives

225

and

400

. Adding them yields

625

. Taking the square root of

625

gives the hypotenuse length of

25

cm. Alternatively, this can be recognized as a scaled

3-4-5

right triangle, where each side is multiplied by

5

(

3 \times 5 = 15

,

4 \times 5 = 20

, and

5 \times 5 = 25

).

Step-by-Step Solution

1

Identify the known values in the Pythagorean theorem formula,

a^2 + b^2 = c^2

.

The legs are

a = 15

and

b = 20

. The hypotenuse

c

is unknown.

The Pythagorean theorem defines the relationship between the lengths of the two legs of a right triangle and its hypotenuse.

2

Substitute the leg values into the theorem and square them.

15^2 + 20^2 = 225 + 400 = 625

. Thus,

c^2 = 625

.

This isolates the square of the unknown hypotenuse (

c^2

) on one side of the equation.

3

Take the square root of both sides to find the hypotenuse.

c = \sqrt{625} = 25

.

The square root gives the actual length of the side rather than its squared value.

Key Concept

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs (

a^2 + b^2 = c^2

).

Question 83Question →

A cross-section of an irrigation canal has the shape of an isosceles trapezoid. The flat bottom of the canal is $10$ meters wide, the open water surface at the top is $20$ meters wide, and the vertical depth of the canal is $12$ meters. To prevent seepage, a waterproof lining is applied only to the flat bottom and the two slanted sides of the canal. What is the total length, in meters, of the canal's cross-section that receives this lining?

Show answer & explanation

Answer: 36

Answer

36

The lined portion consists of the flat bottom and the two slanted sides. By dropping perpendicular altitude lines from the top corners to the bottom plane, two identical right triangles are formed on the left and right edges. The horizontal base of each right triangle is half of the difference between the top and bottom widths:

(20 - 10) / 2 = 5

meters. The vertical height of the canal is given as

12

meters. Using the Pythagorean theorem (

a^2 + b^2 = c^2

), the length of each slanted side is

\sqrt{5^2 + 12^2} = \sqrt{169} = 13

meters. The total length of the waterproof lining is the sum of the bottom width and the two slanted sides:

10 + 13 + 13 = 36

meters.

Step-by-Step Solution

1

Calculate the horizontal difference on each side of the trapezoid.

5 meters

To form a right triangle to calculate the slanted side, we must distribute the difference between the top and bottom widths equally to both sides.

2

Apply the Pythagorean theorem to find the length of the slanted sides.

13 meters

Each slanted side acts as the hypotenuse of a right triangle with legs of 5 meters (horizontal) and 12 meters (vertical height).

3

Sum the lengths of the sections that receive the lining.

36 meters

The lining is only applied to the bottom (10 meters) and the two slanted sides (13 meters each). The top is open and excluded.

Key Concept

Calculating the partial perimeter of a composite/trapezoidal figure using the Pythagorean theorem.

Question 84Question →

A laser cutting machine is programmed to make two straight intersecting cuts on a sheet of acrylic. The measures of a pair of vertical angles created by the intersection are $(9x - 18)^\circ$ and $(5x + 46)^\circ$ . What is the measure of the acute angle formed by these cuts, in degrees?

Show answer & explanation

Answer: 54

Answer

The measure of the acute angle is 54 degrees.

Setting the expressions for the vertical angles equal to each other gives

9x - 18 = 5x + 46

. Solving for

x

yields

x = 16

. Substituting

16

back into the expression

(9x - 18)^\circ

results in a vertical angle of

126^\circ

. Since the intersecting lines form supplementary adjacent angles, the adjacent angle is

180^\circ - 126^\circ = 54^\circ

. This represents the acute angle.

Step-by-Step Solution

1

Set up an equation by equating the two algebraic expressions.

9x - 18 = 5x + 46

Vertical angles formed by intersecting lines are always equal in measure.

2

Solve the equation for the variable

x

.

4x = 64 \Rightarrow x = 16

Isolating

x

provides the numerical value needed to evaluate the angle expressions.

3

Substitute

x = 16

back into either original expression.

9(16) - 18 = 144 - 18 = 126^\circ

Evaluating the expression gives the actual degree measure of the vertical angles.

4

Calculate the measure of the adjacent acute angle.

180^\circ - 126^\circ = 54^\circ

Adjacent angles on a straight line are supplementary (add up to

180^\circ

). Because

54^\circ < 90^\circ

, this is the required acute angle.

Key Concept

Vertical angles are equal, and adjacent angles formed by intersecting lines are supplementary.

Question 85Question →

A cube-shaped block has an edge length of $5$ centimeters. What is the total surface area of the block in square centimeters?

25

125

150

900

Show answer & explanation

Answer: 150

Answer

The total surface area of the cube is 150 square centimeters.

The total surface area of a cube is calculated using the formula

SA = 6s^2

, where

s

is the edge length. Substituting

s = 5

gives

6(5)^2

. By following the order of operations, square the

5

first to get

25

, and then multiply by

6

to find the total surface area of

150

square centimeters.

Step-by-Step Solution

1

Identify the formula for the total surface area of a cube.

SA = 6s^2

A cube has 6 identical square faces, and the area of each face is the edge length squared (

s^2

).

2

Substitute the given edge length (

s = 5

) into the formula.

SA = 6(5)^2

This sets up the equation with the specific dimensions of the block.

3

Evaluate the exponent first, following the order of operations.

5^2 = 25

Exponents must be calculated before multiplication.

4

Multiply the area of one face by the number of faces.

6 \times 25 = 150

This provides the combined area of all 6 faces, which is the total surface area.

Key Concept

Volume and Surface Area of 3D Figures

Estimated Time:45s

Question 86Question →

A city park is shaped like a trapezoid with parallel sides measuring $140$ meters and $100$ meters. The perpendicular distance between these parallel sides is $60$ meters. Inside the park, there is a rectangular concrete plaza that measures $40$ meters by $25$ meters. The rest of the park is covered in grass. What is the total area covered in grass, in square meters?

Show answer & explanation

Answer: 6200

Answer

6200

The total area of the trapezoid is found by multiplying the average of the two bases by the height, resulting in

7200

square meters. The rectangular plaza has an area of

40 \times 25 = 1000

square meters. Subtracting the inner plaza's area from the total area yields

6200

square meters for the grass.

Step-by-Step Solution

1

Calculate the total area of the trapezoidal park.

The area of the park is

7200

square meters.

The formula for the area of a trapezoid is

A = \frac{1}{2}(b_1 + b_2)h

. Substituting the given dimensions yields

A = \frac{1}{2}(140 + 100) \times 60 = 7200

.

2

Calculate the area of the rectangular concrete plaza.

The area of the plaza is

1000

square meters.

The area of a rectangle is length times width. Calculating

40 \times 25

yields

1000

.

3

Subtract the area of the plaza from the total area of the park to find the grass area.

The total area covered in grass is

6200

square meters.

The grass covers the portion of the park that is not the plaza. Subtracting

1000

from

7200

leaves

6200

.

Key Concept

Calculating and subtracting areas of composite 2D figures

Question 87Question →

A structural engineer is designing a roof truss. In the design, two parallel horizontal support beams are intersected by a diagonal brace. The consecutive interior angles (also known as same-side interior angles) formed by this intersection measure $(4x - 15)^\circ$ and $(2x + 45)^\circ$ . What is the measure of the larger of these two angles?

55^\circ

65^\circ

95^\circ

115^\circ

Show answer & explanation

Answer: $95^\circ$

Answer

The measure of the larger angle is

95^\circ

.

Consecutive interior angles formed by two parallel lines and a transversal are supplementary, which means they add up to

180^\circ

. Setting up the equation

(4x - 15) + (2x + 45) = 180

and combining like terms yields

6x + 30 = 180

. Subtracting

30

gives

6x = 150

, so

x = 25

. Substituting

25

back into the expressions gives angles of

85^\circ

and

95^\circ

. Comparing the two, the larger angle is

95^\circ

.

Step-by-Step Solution

1

Identify the geometric relationship between the given angles.

The angles are consecutive (same-side) interior angles.

Consecutive interior angles formed by parallel lines and a transversal are supplementary, meaning their sum is

180^\circ

.

2

Set up an algebraic equation based on the supplementary relationship.

(4x - 15) + (2x + 45) = 180

Adding the expressions for both angles and setting them equal to

180

allows us to solve for

x

.

3

Combine like terms and solve for

x

.

6x + 30 = 180

, so

6x = 150

, giving

x = 25

.

Isolating

x

provides the variable value needed to find the specific angle measures.

4

Substitute

x = 25

back into both original expressions.

First angle:

4(25) - 15 = 100 - 15 = 85^\circ

. Second angle:

2(25) + 45 = 50 + 45 = 95^\circ

.

This step determines the actual degree measures of the two angles.

5

Compare the two calculated angles.

The larger angle is

95^\circ

.

The question asks specifically for the measure of the larger angle.

Key Concept

Same-side (consecutive) interior angles formed by parallel lines cut by a transversal are supplementary (sum to

180^\circ

).

Estimated Time:1m 15s

Question 88Question →

A solid concrete pillar is built in the shape of a rectangular prism. It has a square base with each side measuring $3$ feet, and a total height of $10$ feet. What is the volume of the pillar in cubic feet?

Show answer & explanation

Answer: 90

Answer

90

The volume of a rectangular prism is calculated by multiplying its length, width, and height. Because the base is a square, both the length and the width are

3

feet. Multiplying

3 \times 3 \times 10

yields a total volume of

90

cubic feet.

Step-by-Step Solution

1

Identify the proper formula to calculate the volume of a rectangular prism.

Volume =

l \times w \times h

This is the standard formula used to find the total 3D space occupied by a box-like object.

2

Determine the values of the length, width, and height from the problem description.

Length (

l

) =

3

feet, Width (

w

) =

3

feet, Height (

h

) =

10

feet.

The base is described as a square with a side measuring 3 feet, meaning both the length and width are 3 feet.

3

Multiply the dimensions to calculate the total volume.

3 \times 3 \times 10 = 90

Multiplying the values provides the final volume in cubic units.

Key Concept

The volume of a rectangular prism is found by multiplying its length, width, and height together.

Alternative Method

You can also think of the volume formula as Volume = Area of the Base × height. First, find the area of the square base (

3 \times 3 = 9

square feet). Then, multiply this base area by the height of the pillar (

9 \times 10 = 90

cubic feet).

Estimated Time:45s

Question 89Question →

A circular security zone has a circumference of $16\pi$ meters. What is the area of this zone in square meters?

16\pi

64\pi

128\pi

256\pi

Show answer & explanation

Answer: $64\pi$

Answer

64\pi

To find the area, the radius of the circle must first be determined using the circumference formula (

C = 2\pi r

). Setting the given circumference equal to the formula (

16\pi = 2\pi r

) gives a radius of

8

meters. The area is then calculated using the formula

A = \pi r^2

, which results in

\pi(8)^2 = 64\pi

square meters.

Step-by-Step Solution

1

Identify the formula for circumference to solve for the radius.

C = 2\pi r = 16\pi

The radius is required to calculate the area of the circle.

2

Solve for the radius (

r

).

r = 8

meters

Dividing both sides of the equation by

2\pi

isolates the radius.

3

Apply the area formula for a circle using the found radius.

A = \pi r^2 = \pi(8)^2 = 64\pi

square meters

Substituting the radius (

8

) into the area formula yields the final required value.

Key Concept

Calculating the area of a circle by first deriving its radius from a given circumference.

Estimated Time:45s

Question 90Question →

A municipal park has a flat rectangular section and a semi-circular section attached to one of the rectangle's shorter sides. The overall length of the park, measured from the straight edge of the rectangle to the farthest curved point of the semi-circle, is $200$ meters. The width of the rectangular section is $100$ meters. What is the outer perimeter of the entire park in meters? (Use $\pi \approx 3.14$ )

657

714

557

18,925

Show answer & explanation

Answer: $557$

Answer

The outer perimeter of the park is

557

meters.

The perimeter is the total distance around the outside of the shape. First, find the radius of the semi-circle, which is half of its

100

-meter diameter (

50

m). Subtract this radius from the total

200

-meter length to find the straight edge of the rectangle (

150

m). The outer boundary consists of one

100

-meter straight edge, two

150

-meter straight edges, and the semi-circular arc. The arc length is half the circumference of a full circle (

0.5 \times 3.14 \times 100 = 157

m). Adding these outer edges together gives

100 + 150 + 150 + 157 = 557

meters.

Step-by-Step Solution

1

Identify the diameter and radius of the semi-circle.

Diameter =

100

m, Radius =

50

m

The semi-circle is attached to the

100

-meter wide side of the rectangle, so its diameter matches that width.

2

Determine the length of the rectangular section's straight sides.

200 - 50 = 150

m

The overall length (

200

m) includes both the rectangle's length and the semi-circle's radius (

50

m).

3

Calculate the arc length of the semi-circular section.

\frac{1}{2} \times \pi \times d = 0.5 \times 3.14 \times 100 = 157

m

The outer curved edge is exactly half of a full circle's circumference.

4

Sum the lengths of all outer boundary segments.

100 + 150 + 150 + 157 = 557

m

The total perimeter includes one short straight edge (

100

m), two long straight edges (

150

m each), and the curved arc (

157

m). The internal dashed line is not part of the outer perimeter.

Key Concept

Calculating unknown dimensions and outer perimeters of composite figures.

Estimated Time:1m 0s

Question 91Question →

A guy wire is anchored to the flat ground and attached to a vertical pole, forming a right triangle. The wire is $10$ meters long and attaches to the pole exactly $8$ meters above the ground. What is the horizontal distance from the base of the pole to the anchor point on the ground, in meters?

6

2

18

\sqrt{164}

Show answer & explanation

Answer: $6$

Answer

The correct answer is

6

.

The scenario describes a right triangle where the hypotenuse is

10

and one leg is

8

. Applying the Pythagorean theorem (

a^2 + b^2 = c^2

), we find the unknown leg distance is

\sqrt{10^2 - 8^2} = \sqrt{100 - 64} = \sqrt{36} = 6

.

Step-by-Step Solution

1

Identify the known sides of the right triangle.

The hypotenuse (the wire) is

10

meters, and one vertical leg (the height on the pole) is

8

meters.

Assigning the correct values to the Pythagorean theorem formula ensures the proper calculation path.

2

Set up the Pythagorean theorem equation.

x^2 + 8^2 = 10^2

, where

x

is the unknown horizontal distance.

The Pythagorean theorem relates the lengths of the legs to the hypotenuse in a right triangle.

3

Square the known values and solve for

x^2

.

x^2 + 64 = 100

, which simplifies to

x^2 = 100 - 64 = 36

.

Isolating the squared variable term is necessary to find the missing side length.

4

Take the square root to find the final distance.

x = \sqrt{36} = 6

.

Taking the square root converts the squared area value back into a linear distance.

Key Concept

The Pythagorean theorem (

a^2 + b^2 = c^2

) allows you to find a missing leg of a right triangle when the hypotenuse and the other leg are known.

Question 92Question →

A cabinetmaker is designing custom geometric inlay patterns using various wood veneers. Match each described angle relationship from the design to the correct pair of angle measures.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Match the complementary angles to 40° and 50°; the vertical angles to 75° and 75°; the linear pair to 45° and 135°; and the alternate interior angles to 110° and 110°.

The correct matches require applying fundamental angle definitions: complementary angles sum to 90 degrees, vertical angles are congruent, a linear pair sums to 180 degrees, and alternate interior angles between parallel lines are congruent.

Step-by-Step Solution

1

Calculate the complementary angles.

The angles are

40^\circ

and

50^\circ

.

Complementary angles sum to exactly

90^\circ

. Set up the equation

x + (x + 10) = 90

, which simplifies to

2x = 80

, so

x = 40

.

2

Calculate the vertical angles.

The angles are

75^\circ

and

75^\circ

.

Vertical angles are always equal in measure. Since they are equal and sum to

150^\circ

, you divide

150^\circ

by 2.

3

Calculate the linear pair angles.

The angles are

45^\circ

and

135^\circ

.

Angles in a linear pair sum to

180^\circ

. Set up the equation

x + 3x = 180

, which simplifies to

4x = 180

, so

x = 45

.

4

Determine the alternate interior angles.

Both angles are

110^\circ

.

When parallel lines are intersected by a transversal, alternate interior angles are equal.

Key Concept

Angle Relationships and Algebraic Substitution

Question 93Question →

A trapezoidal wooden sign has a height of $18$ inches. Its parallel top and bottom edges measure $24$ inches and $36$ inches, respectively. If a rectangular hole measuring $8$ inches by $6$ inches is cut out of the sign to mount a digital clock, what is the area of the remaining wooden surface in square inches?

Show answer & explanation

Answer: 492

Answer

492

The remaining area is found by subtracting the area of the inner rectangle from the area of the outer trapezoid. The trapezoid area is

\frac{1}{2}(24 + 36) \times 18 = 540

. The rectangle area is

8 \times 6 = 48

. The difference is

540 - 48 = 492

.

Step-by-Step Solution

1

Calculate the area of the full trapezoidal sign.

540 square inches

This gives the total area before the hole is cut, using the formula 0.5 * (base1 + base2) * height.

2

Calculate the area of the rectangular cutout.

48 square inches

This represents the area that is removed from the sign, calculated as length * width.

3

Subtract the cutout area from the total area.

492 square inches

Subtracting the removed area from the initial total area gives the area of the remaining solid surface.

Key Concept

Area of composite 2D figures involving trapezoids and rectangles.

Estimated Time:1m 0s

Question 94Question →

A sail on a small boat is shaped like a right triangle. The horizontal bottom edge of the sail is $7$ feet long, and the vertical edge is $24$ feet high. What is the length, in feet, of the longest edge (the hypotenuse) of the sail?

Show answer & explanation

Answer: 25

Answer

25

The correct answer is found by applying the Pythagorean Theorem (

a^2 + b^2 = c^2

). Substituting the lengths of the two legs (

7

and

24

), the equation becomes

7^2 + 24^2 = c^2

. Squaring the numbers yields

49 + 576

. Adding these together gives

625 = c^2

. Finally, taking the square root of

625

results in

25

, which represents the length of the hypotenuse.

Step-by-Step Solution

1

Identify the known sides of the right triangle.

Leg

a = 7

feet and leg

b = 24

feet.

The horizontal and vertical edges of the sail represent the two perpendicular legs of a right triangle.

2

Set up the Pythagorean Theorem equation.

7^2 + 24^2 = c^2

The Pythagorean Theorem states that the sum of the squares of the legs is equal to the square of the hypotenuse (

c

).

3

Square the lengths of the legs.

49 + 576 = c^2

To evaluate the equation, the individual lengths must be squared before adding them.

4

Add the squared values together.

625 = c^2

Combining the squared leg lengths yields the square of the hypotenuse.

5

Calculate the square root to solve for c.

c = 25

Taking the square root of

625

gives the final length of the hypotenuse.

Key Concept

Pythagorean Theorem

Question 95Question →

Match each advanced circular geometry scenario with its exact calculated numerical value.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Match the scenarios to their respective values: the wheel makes

5

revolutions, the sector angle is

150^{\circ}

, the track's outer circumference is

50\pi

yards, and the inscribed square's area is

128

square inches.

Each scenario relies on accurately applying distinct circular properties. 1) The distance traveled by a wheel uses the circumference formula (

C=\pi d

). 2) The sector area requires using the proportion of the circle

\frac{\theta}{360}\pi r^2

. 3) Concentric track perimeters require adding the width to the inner radius before recalculating circumference. 4) An inscribed square has a diagonal equal to the circle's diameter, allowing for area calculation via the diagonal.

Step-by-Step Solution

1

Calculate the number of wheel revolutions.

The circumference is

24\pi

. Dividing the total distance

120\pi

by

24\pi

yields

5

revolutions.

One full revolution covers a linear distance equal to exactly one circumference.

2

Determine the sector's central angle.

Using the area formula

A = \frac{\theta}{360}\pi r^2

, setting

15\pi = \frac{\theta}{360}\pi(36)

simplifies to

\theta = 150^{\circ}

.

The area of a sector is a proportional fraction of the full circle's area based on its central angle.

3

Find the outer circumference of the circular track.

The inner radius is

20

(from

40\pi = 2\pi r

). The outer radius is

20 + 5 = 25

. The outer circumference is

2\pi(25) = 50\pi

.

Concentric circles share the same center; the outer radius equals the inner radius plus the track's width.

4

Calculate the area of the inscribed square.

The circle's radius is

8

, so the diameter (and square's diagonal) is

16

. The square's area is

\frac{1}{2}(16)^2 = 128

.

An inscribed square's diagonal is strictly equal to the diameter of the circumscribing circle.

Key Concept

Advanced applications of circle formulas including circumference, sector area, and relationships with inscribed shapes.

Question 96Question →

A circular fountain has a diameter of $12$ meters and is surrounded by a paved walkway that is $3$ meters wide. Which of the following represents the area of the walkway in square meters?

180\pi

81\pi

45\pi

6\pi

Show answer & explanation

Answer: $45\pi$

Answer

The correct answer is

45\pi

.

The correct answer is found by subtracting the area of the inner circle from the area of the larger outer circle. The inner radius is

6

meters (half of the

12

-meter diameter), giving an area of

36\pi

. The outer radius is

9

meters (

6

meters plus the

3

-meter walkway), giving a total area of

81\pi

. The difference is

81\pi - 36\pi = 45\pi

.

Step-by-Step Solution

1

Find the radius of the inner fountain.

12 \div 2 = 6

meters.

The area formula for a circle requires the radius, which is half of the diameter.

2

Calculate the area of the fountain.

Area =

\pi \times 6^2 = 36\pi

square meters.

This establishes the area of the space not covered by the walkway.

3

Find the radius of the total circular area (fountain plus walkway).

6 + 3 = 9

meters.

The walkway adds

3

meters to the radius of the inner circle.

4

Calculate the total area.

Area =

\pi \times 9^2 = 81\pi

square meters.

This gives the combined area of both the fountain and the walkway.

5

Subtract the fountain's area from the total area.

81\pi - 36\pi = 45\pi

square meters.

Subtracting the inner area isolates the area of just the paved walkway.

Key Concept

To find the area of a ring (annulus), calculate the area of the outer circle and subtract the area of the inner circle.

Question 97Question →

Angle $A$ and Angle $B$ are supplementary angles. If the measure of Angle $A$ is $(6x - 10)^\circ$ and the measure of Angle $B$ is $(4x + 30)^\circ$ , what is the measure of Angle $A$ ?

106^\circ

32^\circ

86^\circ

122^\circ

Show answer & explanation

Answer: $86^\circ$

Answer

The measure of Angle

A

is

86^\circ

.

The correct answer is

86^\circ

. Because Angle

A

and Angle

B

are supplementary, their measures add up to

180^\circ

. Setting up the equation

(6x - 10) + (4x + 30) = 180

and combining like terms yields

10x + 20 = 180

. Solving for

x

gives

x = 16

. Substituting

16

into the expression for Angle

A

(

6x - 10

) results in

6(16) - 10 = 86^\circ

.

Step-by-Step Solution

1

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

(6x - 10) + (4x + 30) = 180

Supplementary angles always have a combined sum of

180^\circ

.

2

Combine like terms on the left side of the equation.

10x + 20 = 180

Grouping the

x

terms (

6x + 4x

) and constant terms (

-10 + 30

) simplifies the algebraic expression.

3

Isolate the variable term by subtracting

20

from both sides.

10x = 160

This removes the constant from the variable side, beginning the process of solving for

x

.

4

Divide by

10

to solve for

x

.

x = 16

Dividing both sides by the coefficient isolates

x

.

5

Substitute the value of

x

back into the expression for Angle

A

.

6(16) - 10 = 96 - 10 = 86^\circ

The question asks for the specific measure of Angle

A

, not just the value of the variable

x

.

Key Concept

Supplementary angles are two angles whose measures add up to

180^\circ

. To find unknown measures, set the sum of their algebraic expressions equal to

180

, solve for the variable, and substitute it back into the target expression.

Question 98Question →

A rectangular landing pad is designed so that its length is exactly three times its width. If the total area of the landing pad is $147$ square meters, what is the perimeter of the landing pad in meters?

Show answer & explanation

Answer: 56

Answer

56

By setting the width as

w

and length as

3w

, the area equation is

3w^2 = 147

. Solving for

w

yields a width of

7

meters, making the length

21

meters. The perimeter is the sum of all four sides:

2(21 + 7) = 56

meters.

Step-by-Step Solution

1

Define variables and set up the area equation.

w \times 3w = 147

The area of a rectangle is width times length, and the length is given as three times the width.

2

Solve for the width (

w

).

w^2 = 49 \Rightarrow w = 7

Dividing both sides by

3

isolates

w^2

, and taking the square root gives the width.

3

Calculate the length and the perimeter.

Length =

21

, Perimeter =

56

Length is

3 \times 7

. Perimeter is

2(21 + 7)

.

Key Concept

Calculating the perimeter of a rectangle when given its area and the proportional relationship between its sides.

Question 99Question →

A drone takes off from a launch pad and flies $8$ kilometers due north. It then turns and flies $15$ kilometers due east. What is the straight-line distance from the launch pad to the drone's current position?

13

kilometers

17

kilometers

23

kilometers

60

kilometers

Show answer & explanation

Answer: $17$ kilometers

Answer

The straight-line distance is

17

kilometers.

The drone's path forms a right triangle because north and east are perpendicular directions. The straight-line distance between the launch pad and the current position is the hypotenuse. Using the Pythagorean theorem, we calculate

8^2 + 15^2 = 289

. The square root of

289

is

17

, making the straight-line distance

17

kilometers.

Step-by-Step Solution

1

Identify the shape formed by the drone's flight path.

The path forms a right triangle, with the north and east paths as the legs.

North and east directions are perpendicular to each other, creating a

90^\circ

angle.

2

Set up the Pythagorean theorem equation.

a^2 + b^2 = c^2

, where

a = 8

and

b = 15

.

The straight-line distance from the start to the current position represents the hypotenuse (

c

).

3

Square the lengths of the two legs and add them together.

8^2 + 15^2 = 64 + 225 = 289

.

This calculates the value of

c^2

according to the theorem.

4

Find the square root of the sum to determine the hypotenuse.

\sqrt{289} = 17

.

Taking the square root of

c^2

gives the actual straight-line distance.

Key Concept

The Pythagorean theorem (

a^2 + b^2 = c^2

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

Question 100Question →

A windshield wiper assembly has a total length of $20$ inches from its pivot point to its tip. The innermost $4$ inches of the assembly is a bare metal arm, and the outer $16$ inches is the rubber wiper blade. When turned on, the assembly sweeps back and forth through an angle of $135^\circ$ . What is the total area of the windshield that is wiped by the rubber blade during one sweep? (Leave your answer in terms of $\pi$ )

144\pi

sq in.

96\pi

sq in.

36\pi

sq in.

12\pi

sq in.

Show answer & explanation

Answer: $144\pi$ sq in.

Answer

The correct answer is

144\pi

sq in.

The region wiped by the blade forms a shape called an annulus sector. To find its area, we calculate the area of the larger sector created by the full assembly (radius = 20) and subtract the area of the smaller un-wiped sector created by the metal arm (radius = 4). Both sectors cover a

135^\circ

angle, which is

\frac{3}{8}

of a full circle. Large sector area is

\frac{3}{8}\pi(20^2) = 150\pi

, and small sector area is

\frac{3}{8}\pi(4^2) = 6\pi

. The difference is

144\pi

.

Step-by-Step Solution

1

Identify the geometric shapes involved in the problem.

The total area swept is a large circular sector with a radius of

20

inches. The inner un-wiped area is a smaller circular sector with a radius of

4

inches. The wiped area is the difference between them.

The rubber blade starts

4

inches away from the pivot, leaving a small sector un-wiped.

2

Determine the fraction of a full circle the wiper sweeps.

\frac{135^\circ}{360^\circ} = \frac{3}{8}

of a circle.

Sector area requires knowing what fraction of the total

360^\circ

circle is being used.

3

Calculate the area of the large sector (total sweep).

\text{Area} = \frac{3}{8} \times \pi \times (20)^2 = \frac{3}{8} \times 400\pi = 150\pi \text{ sq in.}

This finds the entire area enclosed by the 20-inch assembly.

4

Calculate the area of the small un-wiped sector.

\text{Area} = \frac{3}{8} \times \pi \times (4)^2 = \frac{3}{8} \times 16\pi = 6\pi \text{ sq in.}

This finds the area swept only by the bare metal arm.

5

Subtract the un-wiped area from the total area.

150\pi - 6\pi = 144\pi \text{ sq in.}

Subtracting gives the area exclusively wiped by the rubber blade.

Key Concept

Sector Area of an Annulus

Geometry

Answer

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