Chapter 4, Problem 9PT

To determine

To calculate: The coordinates of $\Delta ABC\text{ where }A\left(0,2\right),B\left(1.5,-1.5\right)\text{}\text{ and }C\left(-2.5,0\right)$ if perimeter of the triangle is three times the original one.

Answer to Problem 9PT

The vertex matrix form is $\left[\begin{array}{ccc}0& 1.5& -2.5\\ 2& -1.5& 0\end{array}\right]$ .

The new coordinates of $\Delta A\text{'}B\text{'}C\text{'}\text{ are }A\text{'}\left(0,6\right),B\text{'}\left(4.5,-4.5\right)\text{}\text{ and }C\text{'}\left(-7.5,0\right)$ .

Explanation of Solution

Given information:

The coordinates of triangle $\Delta ABC\text{ where }A\left(0,2\right),B\left(1.5,-1.5\right)\text{}\text{ and }C\left(-2.5,0\right)$ .

Calculation:

Consider the coordinates of triangle $\Delta ABC\text{ where }A\left(0,2\right),B\left(1.5,-1.5\right)\text{}\text{ and }C\left(-2.5,0\right)$ .

The vertex matrix form is provided below,

  $\left[\begin{array}{ccc}0& 1.5& -2.5\\ 2& -1.5& 0\end{array}\right]$

According to the question the perimeter of the triangle is three times the original one.

Since, perimeter is the sum of the lengths of the sides of the triangle.

So, if perimeter is to be three times the original one, each coordinate needs to be three times.

For the new coordinates of the triangle the old vertex matrix must be multiplied with 3

Therefore,

  $3\left[\begin{array}{ccc}0& 1.5& -2.5\\ 2& -1.5& 0\end{array}\right]=\left[\begin{array}{ccc}0& 4.5& -7.5\\ 6& -4.5& 0\end{array}\right]$

Now, new coordinates are $A\text{'}\left(0,6\right),B\text{'}\left(4.5,-4.5\right)\text{}\text{ and }C\text{'}\left(-7.5,0\right)$

Thus, the new coordinates of $\Delta A\text{'}B\text{'}C\text{'}\text{ are }A\text{'}\left(0,6\right),B\text{'}\left(4.5,-4.5\right)\text{}\text{ and }C\text{'}\left(-7.5,0\right)$ .