Chapter 4.8, Problem 35HP

To determine

Toproof:

The resulting figure is congruent to the original triangle.

Explanation of Solution

Given information:

Add $n$ units to each $x-$ coordinates of a triangle and $m$ to each $y-$ coordinates.

Proof: Let, $△ABC$ be a triangle with $A\left(0,0\right),B\left(a,0\right),C\left(b,c\right)$ .

Suppose that one adds $n$ units to each $x-$ coordinates of the vertices of $△ABC$ , $m$ to each $y-$ coordinate. The resulting triangle $△DEF$ is with $D\left(n,m\right),E\left(a+n,m\right),F\left(b+n,c+m\right)$ .

Now, first find each length of $△ABC$ . So, length of $\overline{AB}$ with $A\left(0,0\right)$ and $B\left(a,0\right)$

  $\begin{array}{l}AB=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(a-0\right)}^2+{\left(0-0\right)}^2}\\ =\sqrt{a^2+0}\\ =a\end{array}$

Now, find the length of $\overline{AC}$ with $A\left(0,0\right)$ and $C\left(b,c\right)$

  $\begin{array}{l}AB=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(b-0\right)}^2+{\left(c-0\right)}^2}\\ =\sqrt{a^2+c^2}\end{array}$

Now find the length of each side of $△DEF$ .

Now get the length of $\overline{DE}$ with $D\left(n,m\right)$ and $E\left(a+n,m\right)$

  $\begin{array}{l}DE=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(a+n-n\right)}^2+{\left(m-m\right)}^2}\\ =\sqrt{a^2+2}\\ =a\end{array}$

Now, find the length of $\overline{EF}$ with $E\left(a+n,m\right)$ and $F\left(b+n,c+m\right)$

  $\begin{array}{l}EF=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(b+n-\left(a+n\right)\right)}^2+{\left(m-m\right)}^2}\\ =\sqrt{{\left(b-a\right)}^2+c^2}\end{array}$

Find the length of $\overline{DF}$ with $D\left(n,m\right)$ and $F\left(b+n,c+m\right)$

  $\begin{array}{l}DF=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(b+n-n\right)}^2+{\left(c+m-m\right)}^2}\\ =\sqrt{b^2+c^2}\end{array}$

Thus, $\overline{AB}\cong \overline{DE},\overline{BC}\cong \overline{EF}$ and $\overline{AC}\cong \overline{DF}$ . $SSS$ Congruence postulate shows $△ABC\cong △DEF$ .