PSAT Math : Triangles

Study concepts, example questions & explanations for PSAT Math

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Example Questions

Example Question #81 : Sat Mathematics

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:

Correct answer:

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

 

 

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

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 6

\dpi{100} \small 8

\dpi{100} \small 4

\dpi{100} \small 10

\dpi{100} \small 5

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. 

Example Question #71 : Triangles

Screen_shot_2013-03-18_at_10.21.29_pm

 

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

Possible Answers:

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.

Example Question #42 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

1

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

Possible Answers:

6.75\pi

6.5\pi

5\pi

6.25\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

Example Question #502 : Geometry

Find the length of the hypotenuse.

Triangle_4_14_c

Note: This is a right triangle.

Possible Answers:

Correct answer:

Explanation:

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

, 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:

 

 

 

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:

Correct answer:

Explanation:

Use the Pythagorean Theorem: , 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:

Example Question #45 : 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

19 miles

9.43 miles

89 miles

4.36 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

Example Question #44 : 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 :

Now we can start solving for :

The length of the hypotenuse is .

Example Question #51 : 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:

Correct answer:

Explanation:

By the Pythagorean Theorem, if  is the hypotenuse and  and  are the legs, .

Set , the known leg, and rewrite the above as:

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

:

 

:

 

:

 

:

 

:

 

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

Example Question #52 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Figure6

Find the perimeter of the polygon.

Possible Answers:

Correct answer:

Explanation:

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

Figure7

Find the hypotenuse of the right triangle with the Pythagorean Theorem, , where  and  are the legs of the triangle and  is its hypotenuse. 

This is our missing length.

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

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