Chapter 5.5, Problem 21PSB

To determine

Tocalculate: which pairs of sides are congruent.

Answer to Problem 21PSB

  $KI\cong KE$

Explanation of Solution

Given information:

The quadrilateral $KITE$ is a kite.

The lengths of sides,

  $\begin{array}{l}KI=2x+30\\ IT=20+3x\\ TE=\left(\text{not given}\right)\\ EK=6-2x\end{array}$

Formula used:

Pair of adjacent sides of a Kite are equal in length.

Calculation:

Case I:Suppose let $KI=IT$

  $\begin{array}{l}\Rightarrow 2x+30=20+3x\\ \Rightarrow 30-20=3x-2x\\ \Rightarrow x=10\end{array}$

  

On substituting $x=10$ in all the given lengths of sides.

  $\begin{array}{l}KI=2x+30=2\left(10\right)+30=20+30=50\\ IT=20+3x=20+3\left(10\right)=20+30=50\\ EK=6-2x=6-2\left(10\right)=6-20=-14\end{array}$

Note that length of side cannot be negative.

Hence, $KI\ne IT$

Case II: Suppose let $KI=KE$

  $\begin{array}{l}\Rightarrow 2x+30=6-2x\\ \Rightarrow 2x+2x=6-30\\ \Rightarrow 4x=-24\\ \Rightarrow x=-6\end{array}$

  

On substituting $x=-6$ in all the given lengths of sides.

  $\begin{array}{l}KI=2x+30=2\left(-6\right)+30=-12+30=18\\ IT=20+3x=20+3\left(-6\right)=20-18=2\\ EK=6-2x=6-2\left(-2\right)=6+4=10\end{array}$

Observe that all the lengths of sides are positive.

Hence $KI=KE$ is true and then $IT=TE$