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PSAT Math : Triangles

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Example Questions

Example Question #1 : Right Triangles

Triangles

Note: Figure NOT drawn to scale.

Refer to the above figure. Given that $\Delta ABC \sim \Delta DE F$ $\displaystyle \Delta ABC \sim \Delta DE F$, give the area of $\Delta DE F$ $\displaystyle \Delta DE F$.

Possible Answers:

The correct answer is not among the other responses.

243.75 $\displaystyle 243.75$

$\displaystyle 1,828.125$

225 $\displaystyle 225$

1,687.5 $\displaystyle 1,687.5$

Correct answer:

1,687.5 $\displaystyle 1,687.5$

Explanation:

By the Pythagorean Theorem,

$BC = \sqrt{(AC)^{2} - (AB)^{2}}$ $\displaystyle BC = \sqrt{(AC)^{2} - (AB)^{2}}$

$= \sqrt{13^{2}- 5^{2}}$ $\displaystyle = \sqrt{13^{2}- 5^{2}}$

$= \sqrt{169- 25}$ $\displaystyle = \sqrt{169- 25}$

$= \sqrt{144}$ $\displaystyle = \sqrt{144}$

= 12 $\displaystyle = 12$

The similarity ratio of $\Delta DE F$ $\displaystyle \Delta DE F$ to $\Delta ABC$ $\displaystyle \Delta ABC$ is

$\frac{EF}{BC} = \frac{90}{12} = 7.5$ $\displaystyle \frac{EF}{BC} = \frac{90}{12} = 7.5$,

This can be used to find $\displaystyle DE$ :

$\frac{DE}{AB} = 7.5$ $\displaystyle \frac{DE}{AB} = 7.5$

$\frac{DE}{5} = 7.5$ $\displaystyle \frac{DE}{5} = 7.5$

$DE = 5 \cdot 7.5 = 37.5$ $\displaystyle DE = 5 \cdot 7.5 = 37.5$

The area of $\Delta DE F$ $\displaystyle \Delta DE F$ is therefore

$\frac{1}{2} \cdot DE \cdot EF = \frac{1}{2} \cdot 37.5 \cdot 90 =1,687.5$ $\displaystyle \frac{1}{2} \cdot DE \cdot EF = \frac{1}{2} \cdot 37.5 \cdot 90 =1,687.5$

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Example Question #22 : Triangles

Triangles

Note: Figures NOT drawn to scale.

Refer to the above figure. Given that $\Delta ABC \sim \Delta DE F$ $\displaystyle \Delta ABC \sim \Delta DE F$ , evaluate $\displaystyle DE$.

Possible Answers:

$\displaystyle 42$

$52 \frac{1}{2}$ $\displaystyle 52 \frac{1}{2}$

$62\frac{1}{2}$ $\displaystyle 62\frac{1}{2}$

$\displaystyle 35$

$\displaystyle 56$

Correct answer:

$\displaystyle 56$

Explanation:

By the Pythagorean Theorem, since $\overline{AC}$ $\displaystyle \overline{AC}$ is the hypotenuse of a right triangle with legs 6 and 8, its measure is

$AC = \sqrt{6^{2} + 8 ^{2}} = \sqrt{36 + 64 } = \sqrt{100} = 10$ $\displaystyle AC = \sqrt{6^{2} + 8 ^{2}} = \sqrt{36 + 64 } = \sqrt{100} = 10$.

The similarity ratio of $\Delta DE F$ $\displaystyle \Delta DE F$ to $\Delta ABC$ $\displaystyle \Delta ABC$ is

$\frac{DF}{AC} = \frac{70}{10} = 7$ $\displaystyle \frac{DF}{AC} = \frac{70}{10} = 7$.

Likewise,

$\frac{DE}{AB} = 7$ $\displaystyle \frac{DE}{AB} = 7$

$\frac{DE}{8} = 7$ $\displaystyle \frac{DE}{8} = 7$

DE = 56 $\displaystyle DE = 56$

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Example Question #72 : Right Triangles

A right triangle has one side equal to 5 and its hypotenuse equal to 14. Its third side is equal to:

Possible Answers:

171

14.87

13.07

Correct answer:

13.07

Explanation:

The Pythagorean Theorem gives us a² + b² = c² for a right triangle, where c is the hypotenuse and a and b are the smaller sides. Here a is equal to 5 and c is equal to 14, so b² = 14² – 5² = 171. Therefore b is equal to the square root of 171 or approximately 13.07.

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Example Question #73 : Right Triangles

Which of the following could NOT be the lengths of the sides of a right triangle?

Possible Answers:

14, 48, 50

5, 7, 10

5, 12, 13

8, 15, 17

12, 16, 20

Correct answer:

5, 7, 10

Explanation:

We use the Pythagorean Theorem and we calculate that 25 + 49 is not equal to 100.
All of the other answer choices observe the theorem a² + b² = c²

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Example Question #2 : How To Find The Length Of The Side Of A Right Triangle

Which set of sides could make a right triangle?

Possible Answers:

6, 7, 8

9, 12, 15

4, 6, 9

10, 12, 16

Correct answer:

9, 12, 15

Explanation:

By virtue of the Pythagorean Theorem, in a right triangle the sum of the squares of the smaller two sides equals the square of the largest side. Only 9, 12, and 15 fit this rule.

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Example Question #1 : How To Find The Length Of The Side Of A Right Triangle

A right triangle with a base of 12 and hypotenuse of 15 is shown below. Find x.

Screen_shot_2013-03-18_at_10.29.39_pm

Possible Answers:

5.5

4.5

3.5

Correct answer:

Explanation:

Using the Pythagorean Theorem, the height of the right triangle is found to be = √(〖15〗²–〖12〗²) = 9, so x=9 – 5=4

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Example Question #102 : Triangles

A right triangle has sides of 36 and 39(hypotenuse). Find the length of the third side

Possible Answers:

12 √6

33√2

Correct answer:

Explanation:

use the pythagorean theorem:

a² + b² = c² ; a and b are sides, c is the hypotenuse

a² + 1296 = 1521

a² = 225

a = 15

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Example Question #2 : How To Find The Length Of The Side Of A Right Triangle

Bob the Helicopter is at 30,000 ft. above sea level, and as viewed on a map his airport is 40,000 ft. away. If Bob travels in a straight line to his airport at 250 feet per second, how many minutes will it take him to arrive?

Possible Answers:

4 hours and 0 minutes

2 hours and 30 minutes

1 hour and 45 minutes

3 minutes and 50 seconds

3 minutes and 20 seconds

Correct answer:

3 minutes and 20 seconds

Explanation:

Draw a right triangle with a height of 30,000 ft. and a base of 40,000 ft. The hypotenuse, or distance travelled, is then 50,000ft using the Pythagorean Theorem. Then dividing distance by speed will give us time, which is 200 seconds, or 3 minutes and 20 seconds.

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Example Question #3 : How To Find The Length Of The Side Of A Right Triangle

A right triangle has two sides, 9 and x, and a hypotenuse of 15. What is x?

Possible Answers:

Correct answer:

Explanation:

We can use the Pythagorean Theorem to solve for x.

9² + x² = 15²

81 + x² = 225

x² = 144

x = 12

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Example Question #5 : How To Find The Length Of The Side Of A Right Triangle

The area of a right traingle is 42. One of the legs has a length of 12. What is the length of the other leg?

Possible Answers:

$6$ $\displaystyle 6$

$5$ $\displaystyle 5$

$11$ $\displaystyle 11$

$9$ $\displaystyle 9$

$7$ $\displaystyle 7$

Correct answer:

$7$ $\displaystyle 7$

Explanation:

$Area= \frac{1}{2}\times base\times height$ $\displaystyle Area= \frac{1}{2}\times base\times height$

$42=\frac{1}{2}\times base\times 12$ $\displaystyle 42=\frac{1}{2}\times base\times 12$

$42=6\times base$ $\displaystyle 42=6\times base$

$base=7$ $\displaystyle base=7$

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PSAT Math : Triangles

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #1 : Right Triangles

Example Question #22 : Triangles

Example Question #72 : Right Triangles

Example Question #73 : Right Triangles

Example Question #2 : How To Find The Length Of The Side Of A Right Triangle

Example Question #1 : How To Find The Length Of The Side Of A Right Triangle

Example Question #102 : Triangles

Example Question #2 : How To Find The Length Of The Side Of A Right Triangle

Example Question #3 : How To Find The Length Of The Side Of A Right Triangle

Example Question #5 : How To Find The Length Of The Side Of A Right Triangle

All PSAT Math Resources

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