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PSAT Math : Trapezoids

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

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:

$2\sqrt{2}$ $\displaystyle 2\sqrt{2}$

$\sqrt{2}$ $\displaystyle \sqrt{2}$

$\displaystyle 4$

$\displaystyle 2$

$\displaystyle 1$

Correct answer:

$\displaystyle 4$

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.

The answer is 4.

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Example Question #2 : Quadrilaterals

Rectangle_3

Note: Figure NOT drawn to scale.

The white region in the above diagram is a trapezoid. What percent of the above rectangle, rounded to the nearest whole percent, is blue?

Possible Answers:

$85 \%$ $\displaystyle 85 \%$

$65 \%$ $\displaystyle 65 \%$

$80 \%$ $\displaystyle 80 \%$

$70 \%$ $\displaystyle 70 \%$

$75 \%$ $\displaystyle 75 \%$

Correct answer:

$70 \%$ $\displaystyle 70 \%$

Explanation:

The area of the entire rectangle is the product of its length and width, or

$100 \times 50 = 5,000$ $\displaystyle 100 \times 50 = 5,000$ .

The area of the white trapezoid is one half the product of its height and the sum of its base lengths, or

$\frac{1}{2} \times (64+100) \times 18 = 1,476$ $\displaystyle \frac{1}{2} \times (64+100) \times 18 = 1,476$

Therefore, the blue polygon has area

$\displaystyle 5,000 - 1,476= 3,524$ .

This is

$\frac{3,524}{5,000} \times 100 = 70.48 \%$ $\displaystyle \frac{3,524}{5,000} \times 100 = 70.48 \%$ of the rectangle.

Rounded, this is 70%.

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Example Question #3 : Quadrilaterals

Thingy_3

Refer to the above diagram. $\displaystyle BD = 6, DX = 9, CX = 24$ .

Give the area of Quadrilateral ABDC $\displaystyle ABDC$ .

Possible Answers:

108 $\displaystyle 108$

264 $\displaystyle 264$

$\displaystyle 90$

165 $\displaystyle 165$

240 $\displaystyle 240$

Correct answer:

165 $\displaystyle 165$

Explanation:

$\angle C \cong \angle BDX$ $\displaystyle \angle C \cong \angle BDX$ , since both are right; by the Corresponding Angles Theorem, $\overline{AC} || \overline{BD}$ $\displaystyle \overline{AC} || \overline{BD}$ , and Quadrilateral ABDC $\displaystyle ABDC$ is a trapezoid.

By the Angle-Angle Similarity Postulate, since

$\angle C \cong \angle BDX$ $\displaystyle \angle C \cong \angle BDX$

and

$\angle X \cong \angle X$ $\displaystyle \angle X \cong \angle X$ (by reflexivity),

$\Delta ACX \sim \Delta BDX$ $\displaystyle \Delta ACX \sim \Delta BDX$ ,

and since corresponding sides of similar triangles are in proportion,

$\frac{AC}{CX} = \frac{BD}{DX}$ $\displaystyle \frac{AC}{CX} = \frac{BD}{DX}$

$\frac{AC}{24} = \frac{6}{9}$ $\displaystyle \frac{AC}{24} = \frac{6}{9}$

$\frac{AC}{24} \times 24 = \frac{6}{9} \times 24$ $\displaystyle \frac{AC}{24} \times 24 = \frac{6}{9} \times 24$

AC = 16 $\displaystyle AC = 16$ , the larger base of the trapozoid;

The smaller base is BD = 6 $\displaystyle BD = 6$ .

$\displaystyle CD = CX - DX = 24-9 = 15$ , the height of the trapezoid.

The area of the trapezoid is

$A = \frac{1}{2} (b + B)h$ $\displaystyle A = \frac{1}{2} (b + B)h$

$A = \frac{1}{2} (BD + AC) \cdot CD$ $\displaystyle A = \frac{1}{2} (BD + AC) \cdot CD$

$A = \frac{1}{2} (6+ 16) \cdot 15 = \frac{1}{2} (22) \cdot 15 = 165$ $\displaystyle A = \frac{1}{2} (6+ 16) \cdot 15 = \frac{1}{2} (22) \cdot 15 = 165$

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PSAT Math : Trapezoids

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

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

Example Question #2 : Quadrilaterals

Example Question #3 : Quadrilaterals

All PSAT Math Resources

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