### All ACT Math Resources

## 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?

**Possible Answers:**

**Correct answer:**

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:

8^{2} + 15^{2} = x^{2}

64 + 225 = x^{2}

289 = x^{2}

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?

**Possible Answers:**

**Correct answer:**

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 ?

**Possible Answers:**

**Correct answer:**

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

(3*x*)^{2} + (4*x*)^{2} = 10^{2}

9*x*^{2} + 16*x*^{2} = 100

25*x*^{2} = 100

*x*^{2} = 4

*x* = 2

With *x* = 2, each side of the square is 4*x*, 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.

**Possible Answers:**

**Correct answer:**

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.

**Possible Answers:**

**Correct answer:**

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 ?

**Possible Answers:**

**Correct answer:**

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?

**Possible Answers:**

4.36 miles

19 miles

89 miles

9.43 miles

13 miles

**Correct answer:**

9.43 miles

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

Apply the Pythagorean Theorem:

a^{2} + b^{2} = c^{2}

25 + 64 = c^{2}

89 = c^{2 }

c = 9.43 miles

### Example Question #86 : Right Triangles

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

**Possible Answers:**

**Correct answer:**

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?

**Possible Answers:**

**Correct answer:**

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.

**Possible Answers:**

**Correct answer:**

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

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: