Chapter 12, Problem 10CT

To determine

To write: The coordinate proof to show that the triangles created by the keyboard stand are congruent.

Answer to Problem 10CT

The triangles created by the keyboard stand are congruent.

Explanation of Solution

Given:

The given figure is as follows.

  

Calculation:

The coordinates of the vertices of $\Delta PQR$ and $\Delta STR$ are $P\left(3,30\right),Q\left(21,30\right),R\left(12,15\right),S\left(21,0\right)$ , and $T\left(3,0\right)$ .

The objective is to prove $\Delta PQR\cong \Delta STR$ .

The formula to calculate the distance between two points is given by $D=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y{1}_{}\right)}^2}$ .

Find the length of the side $PQ$ .

  $\begin{array}{c}PQ=\sqrt{{\left(21-3\right)}^2+{\left(30-30\right)}^2}\\ =\sqrt{{18}^2}\\ =18\end{array}$

Find the length of the side $QR$ .

  $\begin{array}{c}QR=\sqrt{{\left(12-21\right)}^2+{\left(15-30\right)}^2}\\ =\sqrt{81+225}\\ =3\sqrt{34}\end{array}$

Find the length of the side $PR$ .

  $\begin{array}{c}PR=\sqrt{{\left(12-3\right)}^2+{\left(15-30\right)}^2}\\ =\sqrt{81+225}\\ =3\sqrt{34}\end{array}$

Find the length of the side $RS$ .

  $\begin{array}{c}RS=\sqrt{{\left(21-12\right)}^2+{\left(0-15\right)}^2}\\ =\sqrt{81+225}\\ =3\sqrt{34}\end{array}$

Find the length of the side $ST$ .

  $\begin{array}{c}ST=\sqrt{{\left(3-21\right)}^2+{\left(0-0\right)}^2}\\ =\sqrt{{18}^2}\\ =18\end{array}$

Find the length of the side $RT$ .

  $\begin{array}{c}RT=\sqrt{{\left(3-12\right)}^2+{\left(0-15\right)}^2}\\ =\sqrt{81+225}\\ =3\sqrt{34}\end{array}$

So, $PQ=ST,QR=TR,PR=SR$ .

By definition of congruent, segments $\overline{PQ}=\overline{ST},\overline{QR}=\overline{TR},\overline{PR}=\overline{SR}$ .

By SSS Congruence theorem $\Delta PQR\cong \Delta STR$ .

Therefore, the triangles created by the keyboard stand are congruent.