## 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?      Explanation:

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 a2. Thus, the area of the square given in the problem is s2.

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) = s2

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

(4 + s)(s) = 2s2

Distribute the s on the left.

4s + s2 = 2s2

Subtract s2 from both sides.

4s = s2

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.

### Example Question #1 : How To Find The Area Of A Trapezoid

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

To solve, simply use the formula for the area of a trapezoid. Thus, ### Example Question #2 : Quadrilaterals 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 ?      Explanation:

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 ?      Explanation:

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:        ### All SAT Math Resources 