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ACT Math : How to find the length of the hypotenuse of a right triangle : Pythagorean Theorem

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Example Questions

Example Question #21 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Given the right triangle in the diagram, what is the length of the hypotenuse?

Possible Answers:

$7\ in$ $\displaystyle 7\ in$

$12\ in$ $\displaystyle 12\ in$

$20\ in$ $\displaystyle 20\ in$

$14\ in$ $\displaystyle 14\ in$

$10\ in$ $\displaystyle 10\ in$

Correct answer:

$10\ in$ $\displaystyle 10\ in$

Explanation:

To find the length of the hypotenuse use the Pythagorean Theorem: $a^{2}+b^{2}=c^{2}$ $\displaystyle a^{2}+b^{2}=c^{2}$

Where $\displaystyle a$ and $\displaystyle b$ are the legs of the triangle, and $\displaystyle c$ is the hypotenuse.

$6^{2}+8^{2}=100$ $\displaystyle 6^{2}+8^{2}=100$

$(100)^{\frac{1}2}=10$ $\displaystyle (100)^{\frac{1}2}=10$

The hypotenuse is 10 inches long.

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Example Question #11 : Apply The Pythagorean Theorem To Determine Unknown Side Lengths In Right Triangles: Ccss.Math.Content.8.G.B.7

Righttriangle

Triangle ABC is a right triangle. If the length of side A = 3 inches and C = 5 inches, what is the length of side B?

Possible Answers:

6 inches

1/2 inches

1 inches

4 inches

4.5 inches

Correct answer:

4 inches

Explanation:

Using the Pythagorean Theorem, we know that $a^{2} + b^{2} = c^{2}$ $\displaystyle a^{2} + b^{2} = c^{2}$.

This gives:

$3^{2} + b^{2} = 5^{2}$ $\displaystyle 3^{2} + b^{2} = 5^{2}$

$9 + b^{2} = 25$ $\displaystyle 9 + b^{2} = 25$

Subtracting 9 from both sides of the equation gives:

$b^{2} = 16$ $\displaystyle b^{2} = 16$

b = 4 $\displaystyle b = 4$ inches

Righttriangle

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Example Question #51 : Right Triangles

Righttriangle

Triangle ABC is a right triangle. If the length of side A = 8 inches and B = 11 inches, find the length of the hypoteneuse (to the nearest tenth).

Possible Answers:

184 inches

14.2 inches

185 inches

13.7 inches

13.6 inches

Correct answer:

13.6 inches

Explanation:

Using the Pythagrean Theorem, we know that $a^{2} + b^{2} = c^{2}$ $\displaystyle a^{2} + b^{2} = c^{2}$.

This tells us:

$8^{2} + 11^{2} = C^{2}$ $\displaystyle 8^{2} + 11^{2} = C^{2}$

$64 + 121 = C^{2}$ $\displaystyle 64 + 121 = C^{2}$

$185 = C^{2}$ $\displaystyle 185 = C^{2}$

Taking the square root of both sides, we find that $\displaystyle C = 13.6$ inches

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Example Question #64 : Geometry

Righttriangle

Given:

A = 6 feet

B = 9 feet

What is the length of the hypoteneuse of the triangle (to the nearest tenth)?

Possible Answers:

10.2 feet

10.8 feet

10.6 feet

10.5 feet

10.1 feet

Correct answer:

10.8 feet

Explanation:

Using the Pythagrean Theorem, we know that $a^{2} + b^{2} = c^{2}$ $\displaystyle a^{2} + b^{2} = c^{2}$.

This tells us:

$6^{2} + 9^{2} = C^{2}$ $\displaystyle 6^{2} + 9^{2} = C^{2}$

$36 + 81 = C^{2}$ $\displaystyle 36 + 81 = C^{2}$

$117 = C^{2}$ $\displaystyle 117 = C^{2}$

Taking the square root of both sides, we find that $\displaystyle C = 10.8$

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Example Question #61 : Geometry

Righttriangle

Given:

A = 2 miles

B = 3 miles

What is the length of the hypoteneuse of triangle ABC, to the nearest tenth?

Possible Answers:

3.5 miles

3.2 miles

3.7 miles

3.4 miles

3.6 miles

Correct answer:

3.6 miles

Explanation:

Using the Pythagrean Theorem, we know that $a^{2} + b^{2} = c^{2}$ $\displaystyle a^{2} + b^{2} = c^{2}$.

This tells us:

$2^{2} + 3^{2} = C^{2}$ $\displaystyle 2^{2} + 3^{2} = C^{2}$

$4 + 9 = C^{2}$ $\displaystyle 4 + 9 = C^{2}$

$13 = C^{2}$ $\displaystyle 13 = C^{2}$

Taking the square root of both sides, we find that C = 3.6 $\displaystyle C = 3.6$

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Example Question #71 : Geometry

Given that two sides of a right triangle measure 2 feet and 3 feet, respectively, with a hypoteneuse of x, what is the perimeter of this right triangle (to the nearest tenth)?

Possible Answers:

9.4 feet

18 feet

6.4 feet

8.6 feet

3.6 feet

Correct answer:

8.6 feet

Explanation:

Using the Pythagrean Theorem, we know that $a^{2} + b^{2} = c^{2}$ $\displaystyle a^{2} + b^{2} = c^{2}$.

This tells us:

$2^{2} + 3^{2} = C^{2}$ $\displaystyle 2^{2} + 3^{2} = C^{2}$

$4 + 9 = C^{2}$ $\displaystyle 4 + 9 = C^{2}$

$13 = C^{2}$ $\displaystyle 13 = C^{2}$

Taking the square root of both sides, we find that C = 3.6 $\displaystyle C = 3.6$

To find the perimeter, we add the side lengths together, which gives us that the perimeter is: $\displaystyle 3 + 2 + 3.6 = 8.6$

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Example Question #52 : Right Triangles

$In\;\Delta GHJ, \;what\;is\;the\;length\;of\;\overline{GJ}?$ $\displaystyle In\;\Delta GHJ, \;what\;is\;the\;length\;of\;\overline{GJ}?$

Possible Answers:

$\displaystyle 10$

$\displaystyle 13$

$\displaystyle 12$

$\displaystyle 5$

Correct answer:

$\displaystyle 13$

Explanation:

$Use\;the\;Pythagorean\;Theorem:A^2+B^2=C^2,\;to\; find\;\overline{GJ}.$ $\displaystyle Use\;the\;Pythagorean\;Theorem:A^2+B^2=C^2,\;to\; find\;\overline{GJ}.$

$\displaystyle 5^2+12^2=C^2$

$\displaystyle 25+144=C^2$

169=C^2 $\displaystyle 169=C^2$

$C=\sqrt{169}=13$ $\displaystyle C=\sqrt{169}=13$

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Example Question #21 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Kathy and Jill are travelling from their home to the same destination. Kathy travels due east and then after travelling 6 miles turns and travels 8 miles due north. Jill travels directly from her home to the destination. How miles does Jill travel?

Possible Answers:

$\dpi{100} \small 10\ miles$ $\displaystyle \dpi{100} \small 10\ miles$

$\dpi{100} \small 14\ miles$ $\displaystyle \dpi{100} \small 14\ miles$

$\dpi{100} \small 12\ miles$ $\displaystyle \dpi{100} \small 12\ miles$

$\dpi{100} \small 8\ miles$ $\displaystyle \dpi{100} \small 8\ miles$

$\dpi{100} \small 16\ miles$ $\displaystyle \dpi{100} \small 16\ miles$

Correct answer:

$\dpi{100} \small 10\ miles$ $\displaystyle \dpi{100} \small 10\ miles$

Explanation:

Kathy's path traces the outline of a right triangle with legs of 6 and 8. By using the Pythagorean Theorem

$\dpi{100} \small 6^{2}+8^{2}=x^{2}$ $\displaystyle \dpi{100} \small 6^{2}+8^{2}=x^{2}$

$\dpi{100} \small 36+64=x^{2}$ $\displaystyle \dpi{100} \small 36+64=x^{2}$

$\dpi{100} \small x=10$ $\displaystyle \dpi{100} \small x=10$ miles

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Example Question #51 : Right Triangles

$Which\; of \;the \;following \;is\; NOT\; true\; about \;the \;hypotenuse \;of \;a\; triangle?$ $\displaystyle Which\; of \;the \;following \;is\; NOT\; true\; about \;the \;hypotenuse \;of \;a\; triangle?$

Possible Answers:

$It\;is\;the\;longest\;side.$ $\displaystyle It\;is\;the\;longest\;side.$

$It\;is\;always\;greater\;than\;1.$ $\displaystyle It\;is\;always\;greater\;than\;1.$

$It\;is\;across\;from\;the\;right\;angle.$ $\displaystyle It\;is\;across\;from\;the\;right\;angle.$

$It\;is\;always\;across\;from\;the\;largest\;angle\;in\;the\;triangle.$ $\displaystyle It\;is\;always\;across\;from\;the\;largest\;angle\;in\;the\;triangle.$

Correct answer:

$It\;is\;always\;greater\;than\;1.$ $\displaystyle It\;is\;always\;greater\;than\;1.$

Explanation:

$The\;hypotenuse\;can\;be\;between\;0\;and\;1.$ $\displaystyle The\;hypotenuse\;can\;be\;between\;0\;and\;1.$

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Example Question #31 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

In order to get to work, Jeff leaves home and drives 4 miles due north, then 3 miles due east, followed by 6 miles due north and, finally, 7 miles due east. What is the straight line distance from Jeff’s work to his home?

Possible Answers:

6√2

2√5

10√2

Correct answer:

10√2

Explanation:

Jeff drives a total of 10 miles north and 10 miles east. Using the Pythagorean theorem (a²+b²=c²), the direct route from Jeff’s home to his work can be calculated. 10²+10²=c². 200=c². √200=c. √100√2=c. 10√2=c

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ACT Math : How to find the length of the hypotenuse of a right triangle : Pythagorean Theorem

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #21 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #11 : Apply The Pythagorean Theorem To Determine Unknown Side Lengths In Right Triangles: Ccss.Math.Content.8.G.B.7

Example Question #51 : Right Triangles

Example Question #64 : Geometry

Example Question #61 : Geometry

Example Question #71 : Geometry

Example Question #52 : Right Triangles

Example Question #21 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #51 : Right Triangles

Example Question #31 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

All ACT Math Resources

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