How to find the length of the side of a trapezoid - SAT Math

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Question

The above figure shows Square . is the midpoint of $\overline{AB}$ ; is the midpoint of $\overline{CD}$ ; is the midpoint of $\overline{CN}$ . Construct $\overline{MO}$ .

. Which of the following expresses the length of $\overline{MO}$ in terms of ?

Answer

Construct $\overline{MN}$ as shown in the diagram below:

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

Since is the midpoint of $\overline{CD}$ ,

$CN = \frac{1}{2} CD = \frac{1}{2} t$

Since is the midpoint of $\overline{CN}$ ,

$ON = \frac{1}{2} CN = \frac{1}{2} \cdot \frac{1}{2} t = \frac{1}{4} t$ .

$\bigtriangleup MNO$ is a right triangle, so, by the Pythagorean Theorem,

$MO = \sqrt{(MN)^{2}+(NO)^{2} }$

Substituting:

$MO= \sqrt{t^{2}+\left (\frac{1}{4}t \right)^{2} }$

$MO= \sqrt{t^{2}+ \frac {1}{16} t ^{2} }$

$MO= \sqrt{ \frac {17}{16} t ^{2} }$

Apply the Product of Radicals and Quotient of Radicals Rules:

$MO= \sqrt{ \frac {17}{16} t ^{2} }$

$= \sqrt{ \frac {17}{16} } \cdot \sqrt{t^{2}}$

$= \sqrt{ \frac {17}{16} } t$

$= \frac { \sqrt{17}}{ \sqrt{16}} \cdot t$

$= \frac { \sqrt{17}}{4} t$

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