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ACT Math : Right Triangles

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Example Questions

Example Question #81 : Geometry

An airplane is 8 miles west and 15 miles south of its destination. Approximately how far is the plane from its destination, in miles?

Possible Answers:

$30\ miles$

$17\ miles$

$23\ miles$

$7\ miles$

Correct answer:

$17\ miles$

Explanation:

A right triangle can be drawn between the airplane and its destination.

Destination

15 miles Act_math_170_01 Airplane

8 miles

We can solve for the hypotenuse, x, of the triangle:

8² + 15² = x²

64 + 225 = x²

289 = x²

x = 17 miles

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Example Question #82 : Geometry

An 8-foot-tall tree is perpendicular to the ground and casts a 6-foot shadow. What is the distance, to the nearest foot, from the top of the tree to the end of the shadow?

Possible Answers:

$\dpi{100} \small 4$

$\dpi{100} \small 6$

$\dpi{100} \small 5$

$\dpi{100} \small 10$

$\dpi{100} \small 8$

Correct answer:

$\dpi{100} \small 10$

Explanation:

In order to find the distance from the top of the tree to the end of the shadow, draw a right triangle with the height(tree) labeled as 8 and base(shadow) labeled as 6:

Screen_shot_2013-08-16_at_12.34.40_am

From this diagram, you can see that the distance being asked for is the hypotenuse. From here, you can either use the Pythagorean Theorem:

$\dpi{100} \small a^{2}+b^{2}=c^{2}$

or you can notice that this is simililar to a 3-4-5 triangle. Since the lengths are just increased by a factor of 2, the hypotenuse that is normally 5 would be 10.

8^2+6^2=c^2

64+36=c^2

100=c^2

10=c

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Example Question #83 : Geometry

Screen_shot_2013-03-18_at_10.21.29_pm

In the figure above, ABCD is a square and is three times the length of . What is the area of ABCD ?

Possible Answers:

100

Correct answer:

Explanation:

Assigning the length of ED the value of x, the value of AE will be 3x. That makes the entire side AD equal to 4x. Since the figure is a square, all four sides will be equal to 4x. Also, since the figure is a square, then angle A of triangle ABE is a right angle. That gives triangle ABE sides of 3x, 4x and 10. Using the Pythagorean theorem:

(3x)² + (4x)² = 10²

9x² + 16x² = 100

25x² = 100

x² = 4

x = 2

With x = 2, each side of the square is 4x, or 8. The area of a square is length times width. In this case, that's 8 * 8, which is 64.

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Example Question #84 : Geometry

The hypotenuse is the diameter of the circle. Find the area of the circle above.

Possible Answers:

$5\pi$

$6.25\pi$

$6.75\pi$

$6.5\pi$

$5.5\pi$

Correct answer:

$6.25\pi$

Explanation:

Using the Pythagorean Theorem, we can find the length of the hypotenuse:

$3^{2}+4^{2}=5^{2}$ .

Therefore the hypotenuse has length 5.

The area of the circle is $\pi r^{2}=\pi \cdot 2.5^{2}=6.25\pi$

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Example Question #46 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Find the length of the hypotenuse.

Triangle_4_14_c

Note: This is a right triangle.

Possible Answers:

$2\sqrt{53}$

$\sqrt{91}$

$4\sqrt{53}$

Correct answer:

$2\sqrt{53}$

Explanation:

To find the length of this hypotenuse, we need to use the Pythagorean Theorem:

c^2=a^2+b^2 , where a and b are the legs and c is the hypotenuse.

Here, c is our missing hypotenuse length, a = 4 ,and b = 14.

Plug these values in and solve for c:

c^2=4^2+14^2=16+196=212

$c=\sqrt{212}=\sqrt{4 \cdot 53}=\sqrt{4}\sqrt{53}=2\sqrt{53}$

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Example Question #47 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Side in the triangle below (not to scale) is equal to . Side is equal to . What is the length of side ?

Right_triangle_with_labeled_sides

Possible Answers:

$\sqrt{146}$

$\sqrt{135}$

$\ 146$

Correct answer:

$\sqrt{146}$

Explanation:

Use the Pythagorean Theorem: $a^{2}+b^{2}=c^{2}$ , where a and b are the legs and c is the hypotenuse.

We know and , so we can plug them in to solve for c:

$5^{2}+11^{2}=c^{2}$

$25+121=c^{2}$

$146=c^{2}$

$c=\sqrt{146}$

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Example Question #48 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Dan drives 5 miles north and then 8 miles west to get to school. If he walks, he can take a direct path from his house to the school, cutting down the distance. How long is the path from Dan's house to his school?

Possible Answers:

13 miles

4.36 miles

19 miles

9.43 miles

89 miles

Correct answer:

9.43 miles

Explanation:

We are really looking for the hypotenuse of a triangle that has legs of 5 miles and 8 miles.

Apply the Pythagorean Theorem:

a² + b² = c²

25 + 64 = c²

89 = c²

c = 9.43 miles

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Example Question #45 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

What is the hypotenuse of a right triangle with side lengths and ?

Possible Answers:

Correct answer:

Explanation:

The Pythagorean Theorem states that . This question gives us the values of and , and asks us to solve for .

Take and and plug them into the equation as and :

$12^{2}+16^{2}=c^{2}$

Now we can start solving for :

$144+256=c^{2}$

$400=c^{2}$

$\sqrt{400}=\sqrt{c^{2}}$

20=c

The length of the hypotenuse is .

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Example Question #61 : Triangles

One leg of a triangle measures 12 inches. Which of the following could be the length of the other leg if the hypotenuse is an integer length?

Possible Answers:

$15\ inches$

$4\ inches$

$20\ inches$

$16\ inches$

$12\ inches$

Correct answer:

$16\ inches$

Explanation:

By the Pythagorean Theorem, if is the hypotenuse and and are the legs, $c = \sqrt{a^{2}+b^{2}}}$ .

Set a=12 , the known leg, and rewrite the above as:

$c = \sqrt{12^{2}+b^{2}}}$

$c = \sqrt{144+b^{2}}}$

We can now substitute each of the five choices for ; the one which yields a whole number for is the correct answer choice.

b=4 :

$c = \sqrt{144+4^{2}}} = \sqrt{144+16}} =\sqrt{160}=12.64...$

b=12 :

$c = \sqrt{144+12^{2}}} = \sqrt{144+144}} =\sqrt{288}=16.97...$

b=15 :

$c = \sqrt{144+15^{2}}} = \sqrt{144+225}} =\sqrt{369}=19.20...$

b=16 :

$c = \sqrt{144+16^{2}}} = \sqrt{144+256}} =\sqrt{400}=20$

b=20 :

$c = \sqrt{144+20^{2}}} = \sqrt{144+400}} =\sqrt{544}=23.32...$

The only value of which yields a whole number for the hypotenuse is 16, so this is the one we choose.

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Example Question #62 : Triangles

Find the perimeter of the polygon.

Possible Answers:

Correct answer:

Explanation:

Divide the shape into a rectangle and a right triangle as indicated below.

Find the hypotenuse of the right triangle with the Pythagorean Theorem, a^2 + b^2 = c^2 , where and are the legs of the triangle and is its hypotenuse.

(8)^2+(6)^2 = c^2

100 = c^2

$\sqrt{100} = \sqrt{c^2}$

c =10

This is our missing length.

Now add the sides of the polygon together to find the perimeter:

20 + 10 + 26 + 8 = 64

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ACT Math : Right Triangles

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #81 : Geometry

Example Question #82 : Geometry

Example Question #83 : Geometry

Example Question #84 : Geometry

Example Question #46 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #47 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #48 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #45 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #61 : Triangles

Example Question #62 : Triangles

All ACT Math Resources

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