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SAT Math : Right Triangles

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Example Questions

Example Question #43 : 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:

$\sqrt{135}$

$\sqrt{146}$

$\ 146$

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 #44 : 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:

19 miles

4.36 miles

13 miles

89 miles

9.43 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 #46 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

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:

$16\ inches$

$20\ inches$

$15\ inches$

$12\ inches$

$4\ inches$

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

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

The lengths of the sides of a right triangle are consecutive integers, and the length of the shortest side is . Which of the following expressions could be used to solve for ?

Possible Answers:

(x+2)-(x+1)=x

(x)(x+1)=(x+2)^2

x^2+(x+2)^2=(x+4)^2

x+x-3=x

x^2+(x+1)^2=(x+2)^2

Correct answer:

x^2+(x+1)^2=(x+2)^2

Explanation:

Since the lengths of the sides are consecutive integers and the shortest side is , the three sides are , (x+1) , and (x+2) .

We then use the Pythagorean Theorem:

$\newline a^2+b^2=c^2 \newline x^2+(x+1)^2=(x+2)^2$

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Example Question #12 : Right Triangles

Square PQRS is on the coordinate plane, and each side of the square is parallel to either the -axis or -axis. Point has coordinates $\left ( -2,-1 \right )$ and point has the coordinates $\left ( 3,4 \right )$ .

Quantity A: $5\sqrt{2}$

Quantity B: The distance between points and

Possible Answers:

Quantity A is greater.

The relationship cannot be determined from the information provided.

The two quantities are equal.

Quantity B is greater.

Correct answer:

The two quantities are equal.

Explanation:

To find the distance between points and , split the square into two 45-45-90 triangles and find the hypotenuse. The side ratio of the 45-45-90 triangle is $s:s:s\sqrt{2}$ , so if the sides have a length of 5, the hypotenuse must be $5\sqrt{2}$ .

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Example Question #94 : Plane Geometry

Two sides of a given triangle are both . If one angle of the triangle is a right angle, then what is the measure of the hypotenuse?

Possible Answers:

$5\sqrt{2}$

$7\sqrt{5}$

$2\sqrt{5}$

Correct answer:

$5\sqrt{2}$

Explanation:

If we know two sides are equal to and we know that one of the angles is a right angle, then that means that this must be a Special Right Triangle where the interior angles are $90$^{\circ}$, 45$^{\circ}$, 45$^{\circ}$$ .

With this special triangle, we also know that the measure of the hypotenuse is equal to the measure of one side of the triangle times the square root of that measure.

Since one leg of the triangle is . then the hypotenuse is equal to $5\sqrt{2}$ .

We could also solve this using the Pythagorean Theorem, like so:

$c^2=\sqrt{5^2+5^2}=\sqrt{50}=\sqrt{2\cdot25}=5\sqrt{2}$

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Example Question #61 : Triangles

Justin travels $15\textup{ feet}$ to the east and $20\textup{ feet}$ to the north. How far away from his starting point is he now?

Possible Answers:

$35\textup{ ft}$

$30\textup{ ft}$

$45\textup{ ft}$

$25\textup{ ft}$

$22\textup{ ft}$

Correct answer:

$25\textup{ ft}$

Explanation:

This is solving for the hypotenuse of a triangle. Using the Pythagorean Theorem, which says that a^2+b^2=c^2

15^2 + 20^2 = c^2

225+400=c^2

625=c^2

25=c

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Example Question #51 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

If a 18-foot light pole casts a shadow of 7 feet on even ground, what is the distance from the top of the light pole to the top of it’s shadow?

Possible Answers:

19.31 ft

17.78 ft

25.12 ft

18.36 ft

21.00 ft

Correct answer:

19.31 ft

Explanation:

First, you will want to draw a small diagram to help you:

Lightpole example

After looking at the diagram, it is clear this is a right triangle problem where we can use the Pythagorean Theorem (a² + b² = c² ).

7^2 + 18^2 = c^2

49 + 324= c^2

373 = c^2

$\displaystyle \sqrt{373}= 19.31$

The distance from the top of the light pole to the top of it’s shadow is 19.31 feet.

Process of Elimination Hint: You can immediate eliminate 2 answers based on the properties of right triangles. The hypotenuse has to be longer than either one of the other two sides (must be greater than 18 ft) and less than the length of the other two sides added together (must be less than 25 feet). Therefore, 17.78 feet and 25.12 feet are unreasonable answers.

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SAT Math : Right Triangles

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #43 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #44 : 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 #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 #12 : Right Triangles

Example Question #94 : Plane Geometry

Example Question #61 : Triangles

Example Question #51 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

All SAT Math Resources

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