### All SAT Math Resources

## Example Questions

### Example Question #1 : Trapezoids

A trapezoid has a base of length 4, another base of length *s*, and a height of length *s*. A square has sides of length *s*. What is the value of *s* such that the area of the trapezoid and the area of the square are equal?

**Possible Answers:**

**Correct answer:**

In general, the formula for the area of a trapezoid is (1/2)(*a* + *b*)(*h*), where *a* and *b* are the lengths of the bases, and *h* is the length of the height. Thus, we can write the area for the trapezoid given in the problem as follows:

area of trapezoid = (1/2)(4 + *s*)(*s*)

Similarly, the area of a square with sides of length *a* is given by *a*^{2}. Thus, the area of the square given in the problem is *s*^{2}.

We now can set the area of the trapezoid equal to the area of the square and solve for *s*.

(1/2)(4 + *s*)(*s*) = *s*^{2}

Multiply both sides by 2 to eliminate the 1/2.

(4 + *s*)(*s*) = 2*s*^{2}

Distribute the *s* on the left.

4*s* + *s*^{2} = 2*s*^{2}

Subtract *s*^{2} from both sides.

4*s* = *s*^{2}

Because *s* must be a positive number, we can divide both sides by *s*.

4 = *s*

This means the value of *s* must be 4.

The answer is 4.

### Example Question #1 : Trapezoids

Find the area of a trapezoid given bases of length 1 and 2 and height of 2.

**Possible Answers:**

**Correct answer:**

To solve, simply use the formula for the area of a trapezoid. Thus,

### Example Question #2 : Trapezoids

The above figure shows Square . is the midpoint of ; is the midpoint of ; is the midpoint of . Construct .

If Square has area , what is the area of Quadrilateral ?

**Possible Answers:**

**Correct answer:**

Let be the common sidelength of the square. The area of the square is .

Construct segment . This divides the square into Rectangle and a right triangle. The dimensions of Rectangle are

and

.

The area of Rectangle s the product of these dimensions:

The lengths of the legs of Right are

and

The area of this right triangle is half the product of these lengths, or

This is seen below:

The sum of these areas is the area of Quadrilateral

.

Substituting for , the area is .

### Example Question #3 : Quadrilaterals

The above figure shows Square . is the midpoint of ; is the midpoint of ; is the midpoint of . Construct .

. Which of the following expresses the length of in terms of ?

**Possible Answers:**

**Correct answer:**

Construct as shown in the diagram below:

Quadrilateral is a rectangle, so opposite sides are congruent. Therefore, .

Since is the midpoint of ,

Since is the midpoint of ,

.

is a right triangle, so, by the Pythagorean Theorem,

Substituting:

Apply the Product of Radicals and Quotient of Radicals Rules: