Chapter 9.4, Problem 33PSD

To determine

To calculate:The length of a sideQT .

Answer to Problem 33PSD

The length of a side QTis 16.97.

Explanation of Solution

Given information:

  $RSTV$ is an isosceles trapezoid.

Side $RS=9,$

Side $RV=12,$

Side $ST=18.$

Formula used:

The below theorem is used:

Pythagoras theorem states that “In a right angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides”.

  

In right angle triangle,

  $a^2+b^2\text{= }{c}^2$

Calculation:

  

Draw $\overline{RP}$ and $\overline{VX}$

  $\overline{SP}\cong \overline{XT}$

So, $\overline{SP}\cong \frac{1}{2}(ST-RV)$

  $SP=3$ .

  $\overline{TQ}\perp \overline{SW}$

Right angles are congruent.

  $\angle Q\cong \angle P$

By Reflexive property, we get

  $\angle S\cong \angle S$

Two triangles are similar by AA similarity rule.

  $\Delta SQT\sim \Delta SPR$

  $\begin{array}{l}\frac{SR}{ST}=\frac{SP}{SQ}\\ \frac{9}{18}=\frac{3}{SQ}\\ 9(SQ)=54\\ \therefore SQ=6\end{array}$

Side QT can be calculated by applying Pythagoras Theorem.

In right angled triangle SQT , we get

  $\begin{array}{l}{\left(ST\right)}^2={\left(SQ\right)}^2+{\left(QT\right)}^2\\ $ {18}^2={(6)}^2+{\left(QT\right)}^2${\left(QT\right)}^2=324-36\\ \therefore QT=\sqrt{288}\\ \therefore QT=16.97\end{array}$