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Geometry

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Question 61Question →

Read the geometric scenario below and calculate the missing perimeter. Enter only the numerical value.

Fill in the blanks below

In △PQR\triangle PQR, an altitude PSPS is drawn to the base QRQR, creating two right triangles, △PQS\triangle PQS and △PSR\triangle PSR. If the length of side PQPQ is 1313 units, the length of side PRPR is 1515 units, and the altitude PSPS is 1212 units, the perimeter of the entire △PQR\triangle PQR is units.
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Answer

42
To find the perimeter of △PQR\triangle PQR, we need the lengths of its three outer sides: PQPQ, PRPR, and QRQR. We are given PQ=13PQ = 13 and PR=15PR = 15. The altitude PSPS divides QRQR into two segments, QSQS and SRSR, forming two right triangles. Using the Pythagorean theorem in △PQS\triangle PQS, we find QS=132−122=25=5QS = \sqrt{13^2 - 12^2} = \sqrt{25} = 5. Using the Pythagorean theorem in △PSR\triangle PSR, we find SR=152−122=81=9SR = \sqrt{15^2 - 12^2} = \sqrt{81} = 9. The total base is QR=QS+SR=5+9=14QR = QS + SR = 5 + 9 = 14. The perimeter is the sum of the outer sides: 13+15+14=4213 + 15 + 14 = 42.

Step-by-Step Solution

1
Use the Pythagorean theorem in right △PQS\triangle PQS to find the length of base segment QSQS.
QS=132−122=169−144=25=5QS = \sqrt{13^2 - 12^2} = \sqrt{169 - 144} = \sqrt{25} = 5
The altitude PSPS forms a right angle with the base, making PQPQ the hypotenuse. The missing leg is found by subtracting the square of the altitude from the square of the hypotenuse.
2
Use the Pythagorean theorem in right △PSR\triangle PSR to find the length of base segment SRSR.
SR=152−122=225−144=81=9SR = \sqrt{15^2 - 12^2} = \sqrt{225 - 144} = \sqrt{81} = 9
Similarly, PRPR is the hypotenuse of the second right triangle, so the missing leg is found by subtracting the square of the altitude from the square of this hypotenuse.
3
Calculate the total length of the base QRQR.
QR=QS+SR=5+9=14QR = QS + SR = 5 + 9 = 14
The entire base QRQR is the sum of the two segments created by the altitude.
4
Calculate the perimeter of the full △PQR\triangle PQR.
Perimeter = PQ+PR+QR=13+15+14=42PQ + PR + QR = 13 + 15 + 14 = 42
The perimeter is the total length of the outer boundary of the triangle, which is the sum of its three main sides.

Key Concept

Applying the Pythagorean theorem in multi-step geometric figures to find missing side lengths and perimeter.
Estimated Time:2m 0s
Question 62Question →

The hypotenuse of a right triangle measures 1313 inches, and one of its legs measures 55 inches. What is the length of the other leg?

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Answer: 1212 inches

Answer

The length of the other leg is 1212 inches.
The correct answer is found by properly substituting the hypotenuse (1313) and the known leg (55) into the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2). Rearranging to solve for the missing leg gives a2=132−52a^2 = 13^2 - 5^2, which simplifies to a2=169−25=144a^2 = 169 - 25 = 144. Taking the square root of 144144 yields 1212.

Step-by-Step Solution

1
Set up the Pythagorean theorem equation (a2+b2=c2a^2 + b^2 = c^2) with the given values.
a2+52=132a^2 + 5^2 = 13^2
The Pythagorean theorem relates the two legs (aa and bb) to the hypotenuse (cc) of a right triangle.
2
Square the known numerical values.
a2+25=169a^2 + 25 = 169
Evaluating the exponents is the first operational step in solving the equation.
3
Isolate the variable term a2a^2 by subtracting 2525 from both sides.
a2=144a^2 = 144
This leaves the squared variable isolated on one side of the equation.
4
Take the square root of both sides to find the length of the leg.
a=12a = 12
The square root is the inverse operation of squaring, yielding the final length.

Key Concept

The Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2) can be rearranged to find a missing leg by subtracting the square of the known leg from the square of the hypotenuse (a2=c2−b2a^2 = c^2 - b^2).
Question 63Question →

A computer monitor is shaped like a rectangle. It has a diagonal measurement of 2626 inches and a height of 1010 inches. What is the perimeter of the computer monitor?

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Answer: 6868 inches

Answer

The correct perimeter of the monitor is 6868 inches.
The correct answer is found by first recognizing that the diagonal, height, and width form a right triangle. Using the Pythagorean theorem, the missing width is calculated as the square root of (262−10226^2 - 10^2), which equals 2424 inches. The perimeter of the rectangular monitor is then 22 times the height plus 22 times the width: 2(10)+2(24)=682(10) + 2(24) = 68 inches.

Step-by-Step Solution

1
Identify the right triangle formed by the monitor's dimensions.
The diagonal (2626 inches) is the hypotenuse, and the height (1010 inches) is one leg.
A rectangle's diagonal divides it into two identical right triangles.
2
Use the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2) to find the missing width.
102+b2=26210^2 + b^2 = 26^2 → 100+b2=676100 + b^2 = 676 → b2=576b^2 = 576 → b=24b = 24 inches.
The width is required to calculate the perimeter of the rectangle.
3
Calculate the perimeter of the rectangle.
P=2l+2w=2(10)+2(24)=20+48=68P = 2l + 2w = 2(10) + 2(24) = 20 + 48 = 68 inches.
The perimeter is the total distance around the outside of the monitor.

Key Concept

Triangles and Pythagorean Theorem
Question 64Question →

A circular helipad has a diameter of 2020 meters. What is the area of the helipad in square meters?

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Answer: 100π100\pi

Answer

The correct area is 100π100\pi square meters.
The area of a circle is found using the formula A=πr2A = \pi r^2. Since the given diameter is 2020 meters, the radius is half of that amount, which is 1010 meters. Squaring the radius gives 100100, so the area is 100π100\pi square meters.

Step-by-Step Solution

1
Identify the given information and what needs to be found.
The diameter of the helipad is 2020 meters, and the area needs to be calculated.
Understanding the starting values is required to choose the correct geometric formulas.
2
Calculate the radius of the helipad.
The radius is half of the diameter: r=20÷2=10r = 20 \div 2 = 10 meters.
The standard formula for the area of a circle requires the radius, not the diameter.
3
Apply the area formula for a circle.
A=πr2=π(10)2=100πA = \pi r^2 = \pi (10)^2 = 100\pi square meters.
Squaring the radius and multiplying by π\pi yields the total area.

Key Concept

Calculating the area of a circle from its diameter.
Question 65Question →

A municipal park features a garden in the shape of a trapezoid. The parallel sides of the garden measure 1818 meters and 1212 meters. If the total area of the garden is 150150 square meters, what is the perpendicular distance between the parallel sides?

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Answer: 1010 meters

Answer

The perpendicular distance between the parallel sides is 1010 meters.
The correct answer accurately applies the trapezoid area formula. By setting 150=12(18+12)h150 = \frac{1}{2}(18 + 12)h, the sum of the bases is calculated as 3030. Multiplying by 12\frac{1}{2} results in 1515. Dividing the total area (150150) by 1515 yields a perpendicular height of 1010 meters.

Step-by-Step Solution

1
Identify the formula for the area of a trapezoid.
A=12(b1+b2)hA = \frac{1}{2}(b_1 + b_2)h
This formula relates the known area and bases to the unknown height.
2
Substitute the known values into the formula.
150=12(18+12)h150 = \frac{1}{2}(18 + 12)h
The given area is 150150, and the bases are 1818 and 1212.
3
Simplify the sum inside the parentheses.
150=12(30)h150 = \frac{1}{2}(30)h
Adding the lengths of the bases gives 3030.
4
Multiply 3030 by 12\frac{1}{2}.
150=15h150 = 15h
This isolates the variable term hh.
5
Divide both sides by 1515 to solve for hh.
h=10h = 10
Dividing by 1515 determines the perpendicular height.

Key Concept

Using the area formula of a trapezoid to solve for an unknown dimension.
Question 66Question →

A rectangular parking lot has a perimeter of 200200 yards. The length of the parking lot is 4040 yards greater than its width. What is the area of the parking lot, in square yards?

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Answer: 2100

Answer

2100
The correct answer is derived by first finding the individual dimensions using the perimeter formula and the given relationship (l=w+40l = w + 40), which gives a width of 30 yards and a length of 70 yards. Multiplying these dimensions yields an area of 2100 square yards.

Step-by-Step Solution

1
Write the perimeter formula for a rectangle.
P=2l+2wP = 2l + 2w
The perimeter is the total distance around the rectangle.
2
Translate the given information into equations.
2l+2w=2002l + 2w = 200 and l=w+40l = w + 40
The problem states the perimeter is 200 yards and the length is 40 yards greater than the width.
3
Substitute the expression for length into the perimeter equation and solve for width.
2(w+40)+2w=200⇒2w+80+2w=200⇒4w=120⇒w=302(w + 40) + 2w = 200 \Rightarrow 2w + 80 + 2w = 200 \Rightarrow 4w = 120 \Rightarrow w = 30
Substitution allows us to create an equation with only one variable.
4
Calculate the length.
l=30+40=70l = 30 + 40 = 70
Now that we know the width is 30 yards, we can add 40 to find the length.
5
Calculate the area of the rectangle.
Area=l×w=70×30=2100Area = l \times w = 70 \times 30 = 2100
The area of a rectangle is found by multiplying its length by its width.

Key Concept

Using algebraic equations to find the dimensions of a rectangle from its perimeter, and then calculating its area.
Question 67Question →

A circle is perfectly inscribed within a square that has a perimeter of 4848 centimeters. What is the area of the region located inside the square but outside the circle, in square centimeters?

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Answer: 144−36π144 - 36\pi

Answer

The correct answer is 144−36π144 - 36\pi.
To find the area of the specified region, the area of the inscribed circle must be subtracted from the area of the square. A perimeter of 48 gives the square a side length of 12 and an area of 144. The inscribed circle's diameter matches the square's side length (12), meaning its radius is 6. Its area is calculated as pi times the radius squared, which is 36pi. Subtracting these gives the final area of 144 - 36pi.

Step-by-Step Solution

1
Find the side length of the square.
The side length is 48÷4=1248 \div 4 = 12 cm.
The perimeter of a square is equal to 44 times its side length.
2
Calculate the area of the square.
The area is 12×12=14412 \times 12 = 144 cm 2^2.
The area of a square is the side length squared.
3
Determine the radius of the inscribed circle.
The radius is 12÷2=612 \div 2 = 6 cm.
Because the circle is perfectly inscribed, its diameter equals the side length of the square (1212 cm). The radius is half of the diameter.
4
Calculate the area of the circle.
The area is π×62=36π\pi \times 6^2 = 36\pi cm 2^2.
The formula for the area of a circle is A=πr2A = \pi r^2.
5
Find the area of the region outside the circle but inside the square.
The area is 144−36π144 - 36\pi cm 2^2.
Subtracting the circle's area from the total square's area leaves the remaining corner regions.

Key Concept

Area of composite figures and inscribed shapes
Question 68Question →

An engineer is evaluating the specifications of a circular pipe and recording various measurements. Match each measurement description with the correct geometric term.

Click a left item, then click its matching right item

Items

The total length of the continuous boundary enclosing the pipe's cross-section
The straight-line measurement from the exact center of the pipe to its outer edge
The longest straight distance across the pipe, passing directly through its center
The total amount of two-dimensional flat space enclosed within the pipe's boundary

Matches

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Answer

The boundary length matches to Circumference, the center-to-edge measurement to Radius, the edge-to-edge measurement through the center to Diameter, and the enclosed space to Area.
Each practical measurement perfectly aligns with a fundamental geometric property of a circle. The continuous outer boundary corresponds to the circumference. The distance from the center point to the edge is the radius. The line segment completely crossing the circle through its center is the diameter. Finally, the total two-dimensional region contained inside is the area.

Step-by-Step Solution

1
Analyze the description for the boundary of the circular pipe.
Matches with Circumference.
By definition, the perimeter or outer boundary of a circle is called its circumference.
2
Analyze the description for the measurement from the center to the outer edge.
Matches with Radius.
The radius is the line segment connecting the center point to any point on the outer edge of the circle.
3
Analyze the description for the full width passing through the center.
Matches with Diameter.
The diameter is the longest chord of a circle, spanning from one edge to the other directly through the center.
4
Analyze the description for the enclosed two-dimensional space.
Matches with Area.
Area is the mathematical term for the amount of surface space enclosed by a two-dimensional shape.

Key Concept

Basic terminology and properties of circles
Question 69Question →

A circular crop field has a diameter of 3030 yards. A dirt road that is 55 yards wide is built along the entire outer edge of the field. What is the area of the dirt road in square yards?

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Answer: 175π175\pi

Answer

The area of the dirt road is 175π175\pi square yards.
The dirt road forms a ring (an annulus) around the circular field. To find this area, calculate the area of the total circular space (field plus road) and subtract the area of the field itself. The field's radius is 1515 yards (30÷230 \div 2), making the total outer radius 2020 yards (15+515 + 5). Subtracting the inner area (225π225\pi) from the outer area (400π400\pi) gives the road's area: 175π175\pi square yards.

Step-by-Step Solution

1
Find the radius of the inner circular field.
rinner=302=15r_{inner} = \frac{30}{2} = 15 yards
The area formula for a circle requires the radius, but the diameter of the field is given.
2
Determine the total radius of the entire area, including the road.
router=15+5=20r_{outer} = 15 + 5 = 20 yards
The dirt road adds 55 yards to the outer edge of the field's radius.
3
Calculate the total area enclosed by the outer edge of the road.
Areaouter=π(20)2=400πArea_{outer} = \pi (20)^2 = 400\pi square yards
This represents the total area of the field and the road combined.
4
Calculate the area of just the inner crop field.
Areainner=π(15)2=225πArea_{inner} = \pi (15)^2 = 225\pi square yards
This area must be subtracted to isolate the area of the road.
5
Subtract the inner area from the total outer area.
400π−225π=175π400\pi - 225\pi = 175\pi square yards
The area of the ring (annulus) is the difference between the two concentric circular areas.

Key Concept

Area of an annulus (a ring formed by two concentric circles)
Question 70Question →

A rectangular community garden is 33 times as long as it is wide. If the total area of the garden is 108108 square feet, what is the perimeter of the garden, in feet?

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Answer: 4848

Answer

The perimeter of the garden is 4848 feet.
The correct answer is found by first determining the dimensions of the garden. Since the length is 33 times the width, the area can be represented as (3w)(w)=3w2(3w)(w) = 3w^2. Setting this equal to the given area of 108108 square feet gives 3w2=1083w^2 = 108. Dividing by 33 yields w2=36w^2 = 36, so the width is 66 feet. The length is 3×6=183 \times 6 = 18 feet. The perimeter formula for a rectangle is P=2(length+width)P = 2(length + width). Substituting the dimensions gives P=2(18+6)=2(24)=48P = 2(18 + 6) = 2(24) = 48 feet.

Step-by-Step Solution

1
Let the width of the garden be ww. The length is 33 times the width, so the length is 3w3w.
Width = ww, Length = 3w3w
Translating the word problem into algebraic expressions allows us to set up an equation.
2
Set up an equation using the formula for the area of a rectangle (Area=length×widthArea = length \times width).
3w×w=1083w \times w = 108
We are given the total area, which links our variable expressions to a known value.
3
Solve the equation for ww.
3w2=108⇒w2=36⇒w=63w^2 = 108 \Rightarrow w^2 = 36 \Rightarrow w = 6
Isolating the variable determines the numerical value for the width.
4
Substitute w=6w = 6 back into the expression for length (3w3w).
Length = 3(6)=183(6) = 18 feet
Both dimensions are needed to calculate the perimeter.
5
Calculate the perimeter using the formula P=2(length+width)P = 2(length + width).
P=2(18+6)=2(24)=48P = 2(18 + 6) = 2(24) = 48 feet
The perimeter represents the total distance around the outside of the rectangular garden.

Key Concept

Perimeter and Area of 2D Figures
Estimated Time:1m 0s
Question 71Question →

A theater crew is building a wooden stage prop in the shape of a right triangle. The vertical height of the prop is 1212 feet and the horizontal base is 55 feet. What is the length of the slanted top edge of the prop?

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Answer: 1313 feet

Answer

The length of the slanted top edge is 1313 feet.
The slanted edge represents the hypotenuse of a right triangle. According to the Pythagorean theorem, the square of the hypotenuse equals the sum of the squares of the two perpendicular legs (a2+b2=c2a^2 + b^2 = c^2). Substituting the given legs (55 and 1212), we calculate 52+122=25+144=1695^2 + 12^2 = 25 + 144 = 169. Taking the square root of 169169 yields 1313 feet.

Step-by-Step Solution

1
Identify the given values as the legs of a right triangle.
The legs are a=5a = 5 and b=12b = 12. The unknown slanted edge is the hypotenuse, cc.
The Pythagorean theorem relates the two perpendicular legs to the longest side.
2
Substitute the leg values into the Pythagorean theorem formula, a2+b2=c2a^2 + b^2 = c^2.
52+122=c25^2 + 12^2 = c^2
This sets up the equation to solve for the missing hypotenuse.
3
Square the numbers and add them together.
25+144=c225 + 144 = c^2, which simplifies to 169=c2169 = c^2.
This isolates c2c^2 on one side of the equation.
4
Take the square root of both sides to find the value of cc.
c=13c = 13
Taking the square root cancels the exponent and gives the final length of the edge.

Key Concept

Using the Pythagorean theorem to calculate the hypotenuse of a right triangle.
Question 72Question →

A circular hatch on a maritime vessel has a diameter of 1414 inches. What is the area of the hatch in square inches?

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Answer: 49π49\pi

Answer

The correct area is 49π49\pi square inches.
The correct answer is found by first determining the radius, which is half of the given diameter (14÷2=714 \div 2 = 7 inches). Applying the area formula A=πr2A = \pi r^2 yields π×72=49π\pi \times 7^2 = 49\pi square inches.

Step-by-Step Solution

1
Determine the radius of the circular hatch.
Radius = 14÷2=714 \div 2 = 7 inches.
The formula for the area of a circle requires the radius, which is exactly half of the diameter.
2
Apply the area formula for a circle.
Area = π×72=49π\pi \times 7^2 = 49\pi square inches.
The area of a circle is found using the formula A=πr2A = \pi r^2.

Key Concept

Calculating the area of a circle by first deriving the radius from the diameter.
Question 73Question →

Match each circular geometry scenario with its corresponding simplified algebraic expression.

Click a left item, then click its matching right item

Items

The area of a circular pathway of uniform width xx surrounding a pool with radius rr
The total perimeter of a semicircular window with radius rr
The number of full revolutions a tire with diameter dd makes to travel a distance DD
The radius of a circle whose area is exactly half the area of a circle with radius RR

Matches

Show answer & explanation

Answer

The correct pairings connect each geometric scenario to its simplified algebraic derivation by applying circle formulas, expanding binomials, and rationalizing denominators.
Each expression is correctly derived by applying the fundamental formulas for circle area and circumference, followed by rigorous algebraic simplification such as factoring, binomial expansion, or radical rationalization.

Step-by-Step Solution

1
Analyze the annular pathway area scenario.
πx(2r+x)\pi x(2r + x)
Requires subtracting the inner pool area from the total outer area and expanding the squared binomial (r+x)2(r+x)^2.
2
Determine the semicircular perimeter expression.
r(π+2)r(\pi + 2)
Requires adding the semicircular arc length to the straight diameter base and factoring out the radius.
3
Calculate the algebraic expression for tire revolutions.
Dπd\frac{D}{\pi d}
Requires dividing the total linear distance by the circumference of the tire, utilizing the diameter.
4
Solve for the half-area radius.
R22\frac{R\sqrt{2}}{2}
Requires setting up an area ratio equation, solving for the new radius, and rationalizing the denominator of the resulting square root.

Key Concept

Algebraic representation and derivation of advanced circular measurements
Question 74Question →

A rectangular garden gate needs a diagonal wooden brace for support. The gate is 44 feet tall and 33 feet wide. What is the length, in feet, of the diagonal brace?

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Answer: 5

Answer

5
The diagonal brace forms the hypotenuse of a right triangle where the width (33 feet) and height (44 feet) are the legs. According to the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2), we substitute the given lengths: 32+42=c23^2 + 4^2 = c^2. This simplifies to 9+16=259 + 16 = 25. Taking the square root of 2525 gives a diagonal length of 55 feet. This forms a common 3−4−53-4-5 Pythagorean triple.

Step-by-Step Solution

1
Identify the right triangle formed by the gate
The legs of the right triangle are 33 and 44, and the hypotenuse is the unknown diagonal brace.
The corners of a rectangular gate form right angles, so drawing a diagonal creates a right triangle.
2
Apply the Pythagorean theorem
32+42=c23^2 + 4^2 = c^2
The Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2) is used to find the missing side length of a right triangle.
3
Calculate the squares of the legs
9+16=c29 + 16 = c^2
Squaring the given lengths prepares the equation for addition.
4
Solve for the hypotenuse
25=c225 = c^2, which means c=5c = 5
Taking the square root of the sum yields the length of the diagonal brace.

Key Concept

Using the Pythagorean theorem to find the hypotenuse of a right triangle.
Question 75Question →

A piece of wire is bent to form a square with an area of 144144 square centimeters. If the same wire is straightened and then bent to form a rectangle whose length is 44 centimeters longer than its width, what is the area of this new rectangle?

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Answer: 140140 square centimeters

Answer

140140 square centimeters
First, find the side length of the square. Since the area is 144144, the side length is 144=12\sqrt{144} = 12. The total length of the wire is the perimeter of the square, which is 12×4=4812 \times 4 = 48. This wire is then used to form a rectangle, so the rectangle's perimeter is also 4848. Let the width of the rectangle be WW. The length is W+4W + 4. The perimeter formula is 2(L+W)=P2(L + W) = P. Substituting the values gives 2(W+W+4)=482(W + W + 4) = 48, which simplifies to 4W+8=484W + 8 = 48. Solving for WW yields 4W=404W = 40, so W=10W = 10. The length is 10+4=1410 + 4 = 14. The area of the new rectangle is 10×14=14010 \times 14 = 140 square centimeters.

Step-by-Step Solution

1
Calculate the total length of the wire using the square's area.
The side of the square is 144=12\sqrt{144} = 12 cm. The perimeter is 12×4=4812 \times 4 = 48 cm.
The total length of the wire remains constant and equals the perimeter of both shapes.
2
Set up an equation for the rectangle's perimeter.
Let width be WW. Length is W+4W + 4. The perimeter is 2(W+(W+4))=482(W + (W + 4)) = 48.
The rectangle is made from the same wire, so its perimeter is 4848 cm, and its length is 44 cm longer than its width.
3
Solve for the width (WW) and length (LL) of the rectangle.
2(2W+4)=48⇒4W+8=48⇒4W=40⇒W=102(2W + 4) = 48 \Rightarrow 4W + 8 = 48 \Rightarrow 4W = 40 \Rightarrow W = 10 cm. The length is 10+4=1410 + 4 = 14 cm.
Solving the linear equation provides the dimensions needed to find the area.
4
Calculate the area of the rectangle.
Area = 10×14=14010 \times 14 = 140 square centimeters.
The area of a rectangle is its length multiplied by its width.

Key Concept

Perimeter and Area of 2D Figures
Question 76Question →

A rectangular shipping crate has a length of 66 feet, a width of 44 feet, and a height of 33 feet. What is the volume of the crate in cubic feet?

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Answer: 72

Answer

72
The volume of a rectangular prism is calculated by multiplying its length, width, and height. Multiplying 66 feet by 44 feet by 33 feet results in exactly 7272 cubic feet.

Step-by-Step Solution

1
Recall the volume formula for a rectangular prism.
V=l×w×hV = l \times w \times h
To find the volume, we must multiply the length, width, and height of the 3D figure.
2
Substitute the given dimensions into the formula.
V=6×4×3V = 6 \times 4 \times 3
The problem states the length is 66 feet, the width is 44 feet, and the height is 33 feet.
3
Multiply the values to find the total volume.
V=24×3=72V = 24 \times 3 = 72
Completing the arithmetic gives the final volume in cubic feet.

Key Concept

Volume of a Rectangular Prism
Question 77Question →

A kite designer is creating a kite in the shape of a rhombus. The two crossing wooden dowels that form the internal diagonals of the kite measure 1616 inches and 3030 inches. What is the total perimeter of the kite's outer edge, in inches?

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Answer: 68

Answer

68
The correct answer is found by recognizing that a rhombus's diagonals bisect each other perpendicularly. This allows the use of the Pythagorean theorem with half-diagonal lengths (88 and 1515) to find a side length of 1717. Multiplying this single side length by the four equal sides of the rhombus yields a total perimeter of 6868 inches.

Step-by-Step Solution

1
Determine the properties of the diagonals in a rhombus.
The diagonals bisect each other at right angles, forming four identical right triangles.
This breaks down the rhombus into simpler geometric shapes (right triangles), allowing for side length calculation.
2
Calculate the lengths of the legs for one of the right triangles.
The legs are 88 inches (16÷216 \div 2) and 1515 inches (30÷230 \div 2).
The diagonals cut each other exactly in half at the intersection.
3
Use the Pythagorean theorem to find the length of the rhombus's outer edge.
The side length is 82+152=64+225=289=17\sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17 inches.
The outer edge forms the hypotenuse of the right triangle created by the half-diagonals.
4
Calculate the total perimeter.
17×4=6817 \times 4 = 68 inches.
A rhombus has four equal sides, so the total perimeter is four times the length of one side.

Key Concept

Perimeter of a Rhombus and Pythagorean Theorem
Question 78Question →

A cylindrical canister has a radius of 33 inches and a height of 55 inches. Which of the following represents the total volume of the canister in cubic inches?

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Answer: 45π45\pi

Answer

45π45\pi
The correct answer is 45π45\pi. The volume of a cylinder is found using the formula V=πr2hV = \pi r^2 h. Squaring the given radius of 33 results in 99. Multiplying 99 by the given height of 55 yields 45π45\pi cubic inches.

Step-by-Step Solution

1
Identify the correct formula for the volume of a cylinder.
V=πr2hV = \pi r^2 h
The question asks for the volume of a cylindrical shape.
2
Substitute the given values into the formula.
V=π(3)2(5)V = \pi (3)^2 (5)
The radius (r) is 3 inches and the height (h) is 5 inches.
3
Square the radius according to the order of operations.
32=93^2 = 9
Exponents must be evaluated before multiplication.
4
Multiply the squared radius by the height and pi.
9×5×π=45π9 \times 5 \times \pi = 45\pi
This completes the volume calculation.

Key Concept

Volume of a Cylinder
Question 79Question →

In a right triangle, the two shorter sides (legs) measure 1010 inches and 2424 inches. What is the length of the hypotenuse in inches?

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Answer: 26

Answer

The length of the hypotenuse is 26 inches.
The hypotenuse is found by applying the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2). When you square the given legs (1010 and 2424), you get 100100 and 576576. Adding these gives 676676. Finally, taking the square root of 676676 results in 2626. This is also recognized as a scaled version of the fundamental 5-12-13 Pythagorean triple (multiplied by 2).

Step-by-Step Solution

1
Recall the Pythagorean theorem formula.
a2+b2=c2a^2 + b^2 = c^2
This formula connects the lengths of the legs (aa and bb) to the hypotenuse (cc) in a right triangle.
2
Substitute the given leg lengths into the formula.
102+242=c210^2 + 24^2 = c^2
The lengths of the two shorter sides are given as 10 and 24.
3
Square the lengths of the legs.
100+576=c2100 + 576 = c^2
Following the order of operations, evaluate the exponents before adding.
4
Add the squared values together.
676=c2676 = c^2
Combine the values to find the square of the hypotenuse.
5
Take the square root of both sides to solve for c.
c=676=26c = \sqrt{676} = 26
Taking the square root provides the final length of the hypotenuse.

Key Concept

Pythagorean Theorem
Question 80Question →

A square metal plate has an area of 144144 square inches. A rectangular metal plate is cut so that it has the exact same width as the square plate, but its length is twice the width of the square. What is the perimeter of the rectangular plate in inches?

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Answer: 72

Answer

72
The area of the square is 144144 square inches, which means each side of the square is 1212 inches. The new rectangular plate has the same width as the square (1212 inches) and a length that is twice that width (2424 inches). To find the perimeter of the rectangle, add the length and width (12+24=3612 + 24 = 36) and multiply by 22 to account for all four sides, resulting in 7272 inches.

Step-by-Step Solution

1
Find the side length of the original square metal plate.
Side = 1212 inches.
The area of a square is the side length squared (s2=144s^2 = 144), so the side is the square root of 144144.
2
Determine the dimensions of the rectangular plate.
Width = 1212 inches, Length = 2424 inches.
The problem states the rectangle's width equals the square's side (1212), and its length is twice that amount (2×12=242 \times 12 = 24).
3
Calculate the perimeter of the rectangular plate.
Perimeter = 2×(12+24)=722 \times (12 + 24) = 72 inches.
The perimeter of a rectangle is found by adding the length and width, then multiplying by 22.

Key Concept

Perimeter and Area of 2D Figures
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