# ACT Math : Triangles

## Example Questions

### Example Question #39 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

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

Explanation:

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

Destination

15 miles    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 #40 : 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?

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:

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

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 #61 : Triangles

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

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 #82 : Right Triangles

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

Explanation:

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

.

Therefore the hypotenuse has length 5.

The area of the circle is

### Example Question #62 : Triangles

Find the length of the hypotenuse.

Note: This is a right triangle.

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 #84 : Right Triangles

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

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 #85 : Right Triangles

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?

4.36 miles

19 miles

89 miles

9.43 miles

13 miles

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 #86 : Right Triangles

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

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 #82 : Sat Mathematics

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?

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 #88 : Right Triangles

Find the perimeter of the polygon.