### All PSAT Math Resources

## Example Questions

### Example Question #421 : Geometry

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 #51 : 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 #41 : 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 ?

**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 #62 : 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:**

89 miles

13 miles

19 miles

4.36 miles

9.43 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 #121 : Geometry

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 #41 : 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:**

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

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:

### Example Question #91 : Geometry

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:**

**Correct answer:**

Since the lengths of the sides are consecutive integers and the shortest side is , the three sides are , , and .

We then use the Pythagorean Theorem:

### Example Question #111 : Geometry

Square is on the coordinate plane, and each side of the square is parallel to either the -axis or -axis. Point has coordinates and point has the coordinates .

Quantity A:

Quantity B: The distance between points and

**Possible Answers:**

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined from the information provided.

**Correct answer:**

The two quantities are equal.

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 , so if the sides have a length of 5, the hypotenuse must be .

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

Justin travels to the east and to the north. How far away from his starting point is he now?

**Possible Answers:**

**Correct answer:**

This is solving for the hypotenuse of a triangle. Using the Pythagorean Theorem, which says that

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