Chapter 7.6, Problem 18PPS

To determine

Find the missing coordinate using given diagram.

Answer to Problem 18PPS

The coordinate of the $Z$ vertexis $\left(12,0\right)$ .

Explanation of Solution

Given:

The given diagram is

  

  $\Delta ABC\sim \Delta AYZ$

Calculation:

Find the distance (using distance formula) of corresponding sides and find the ratio.

  $\begin{array}{l}AY=\sqrt{{\left(0-0\right)}^2+{\left(0-8\right)}^2}=\sqrt{64}=8\\ AB=\sqrt{{\left(0-0\right)}^2+{\left(0-4\right)}^2}=\sqrt{16}=4\end{array}$

  $\frac{AY}{AB}=\frac{8}{4}=2$

  $\begin{array}{l}AZ=\sqrt{{\left(0-p\right)}^2+{\left(0-q\right)}^2}=\sqrt{p^2+q^2}\\ AC=\sqrt{{\left(0-6\right)}^2+{\left(0-0\right)}^2}=\sqrt{36}=6\end{array}$

  $\frac{AZ}{AC}=\frac{p}{6}$

Given $\Delta ABC\sim \Delta AYZ$

So, the length of sides are proportional.

  $\begin{array}{l}\frac{AY}{AB}=\frac{AZ}{AC}\\ 2=\frac{AZ}{6}\\ \Rightarrow AZ=12\end{array}$

In the given figure,

y-coordinate of the $Z$ vertex $=0$

If the y-coordinate of the $Z$ vertex $=0$ , then thex-coordinate of the $Z$ vertex $=12$

Hence thecoordinate of the $Z$ vertexis $\left(12,0\right)$ .