### All PSAT Math Resources

## Example Questions

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

When the side of a certain square is increased by 2 inches, the area of the resulting square is 64 sq. inches greater than the original square. What is the length of the side of the original square, in inches?

**Possible Answers:**

17

14

15

18

16

**Correct answer:**

15

Let x represent the length of the original square in inches. Thus the area of the original square is x^{2}. Two inches are added to x, which is represented by x+2. The area of the resulting square is (x+2)^{2}. We are given that the new square is 64 sq. inches greater than the original. Therefore we can write the algebraic expression:

x^{2} + 64 = (x+2)^{2}

FOIL the right side of the equation.

x^{2} + 64 = x^{2} + 4x + 4

Subtract x^{2 }from both sides and then continue with the alegbra.

64 = 4x + 4

64 = 4(x + 1)

16 = x + 1

15 = x

Therefore, the length of the original square is 15 inches.

If you plug in the answer choices, you would need to add 2 inches to the value of the answer choice and then take the difference of two squares. The choice with 15 would be correct because 17^{2} -15^{2} = 64.

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

If the area of a square is 25 inches squared, what is the perimeter?

**Possible Answers:**

25

15

10

20

Not enough information

**Correct answer:**

20

The area of a square is equal to length times width or length squared (since length and width are equal on a square). Therefore, the length of one side is or The perimeter of a square is the sum of the length of all 4 sides or

### Example Question #1 : Rectangles

What is the length of the diagonal of a rectangle that is 3 feet long and 4 feet wide?

**Possible Answers:**

**Correct answer:**

The diagonal of the rectangle is equivalent to finding the length of the hypotenuse of a right triangle with sides 3 and 4. Using the Pythagorean Theorem:

Therefore the diagonal of the rectangle is 5 feet.

### Example Question #1 : How To Find The Length Of The Diagonal Of A Rectangle

The length and width of a rectangle are in the ratio of 3:4. If the rectangle has an area of 108 square centimeters, what is the length of the diagonal?

**Possible Answers:**

9 centimeters

24 centimeters

18 centimeters

12 centimeters

15 centimeters

**Correct answer:**

15 centimeters

The length and width of the rectangle are in a ratio of 3:4, so the sides can be written as 3*x* and 4*x*.

We also know the area, so we write an equation and solve for x:

(3*x*)(4*x*) = 12*x*^{2 }= 108.

x^{2} = 9

*x* = 3

Now we can recalculate the length and the width:

length = 3x = 3(3) = 9 centimeters

width = 4x = 4(3) = 12 centimeters

Using the Pythagorean Theorem we can find the diagonal, *c*:

length^{2} + width^{2} = c^{2}

9^{2} + 12^{2 }= *c*^{2}

81 + 144 = c^{2}

225 = c^{2}

*c *= 15 centimeters

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

The two rectangles shown below are similar. What is the length of EF?

**Possible Answers:**

10

8

6

5

**Correct answer:**

10

When two polygons are similar, the lengths of their corresponding sides are proportional to each other. In this diagram, AC and EG are corresponding sides and AB and EF are corresponding sides.

To solve this question, you can therefore write a proportion:

AC/EG = AB/EF ≥ 3/6 = 5/EF

From this proportion, we know that side EF is equal to 10.

### Example Question #42 : Quadrilaterals

A rectangle is x inches long and 3x inches wide. If the area of the rectangle is 108, what is the value of x?

**Possible Answers:**

3

6

12

4

8

**Correct answer:**

6

Solve for x

Area of a rectangle A = lw = x(3x) = 3x^{2} = 108

x^{2} = 36

x = 6

### Example Question #43 : Quadrilaterals

If the area of rectangle is 52 meters squared and the perimeter of the same rectangle is 34 meters. What is the length of the larger side of the rectangle if the sides are integers?

**Possible Answers:**

13 meters

14 meters

15 meters

12 meters

16 meters

**Correct answer:**

13 meters

Area of a rectangle is = lw

Perimeter = 2(l+w)

We are given 34 = 2(l+w) or 17 = (l+w)

possible combinations of l + w

are 1+16, 2+15, 3+14, 4+13... ect

We are also given the area of the rectangle is 52 meters squared.

Do any of the above combinations when multiplied together= 52 meters squared? yes 4x13 = 52

Therefore the longest side of the rectangle is 13 meters

### Example Question #6 : Rectangles

Note: Figure NOT drawn to scale.

In the above figure,

.

.

Give the perimeter of .

**Possible Answers:**

**Correct answer:**

We can use the Pythagorean Theorem to find :

The similarity ratio of to is

so multiplied by the length of a side of is the length of the corresponding side of . We can subsequently multiply the perimeter of the former by to get that of the latter:

### Example Question #7 : Rectangles

Note: Figure NOT drawn to scale.

In the above figure,

.

.

Give the area of .

**Possible Answers:**

Insufficient information is given to determine the area.

**Correct answer:**

Corresponding sidelengths of similar polygons are in proportion, so

, so

We can use the Pythagorean Theorem to find :

The area of is

### Example Question #1 : How To Find If Rectangles Are Similar

Note: Figure NOT drawn to scale.

In the above figure,

.

.

Give the area of Polygon .

**Possible Answers:**

**Correct answer:**

Polygon can be seen as a composite of right and , so we calculate the individual areas and add them.

The area of is half the product of legs and :

Now we find the area of . We can do this by first finding using the Pythagorean Theorem:

The similarity of to implies

so

The area of is the product of and :

Now add: , the correct response.

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