Chapter 4, Problem 11PT

To determine

To calculate: The coordinates of $\Delta XYZ\text{ where }X\left(1,2\right),Y\left(3,6\right)\text{}\text{ and }Z\left(-1,4\right)$ if perimeter of the triangle is to doubled.

Answer to Problem 11PT

The new coordinates of $\Delta X\text{'}Y\text{'}Z\text{'}\text{ are }X\text{'}\left(2,4\right),Y\text{'}\left(6,12\right)\text{}\text{ and }Z\text{'}\left(-2,8\right)$ .

Explanation of Solution

Given information:

The coordinates of triangle $\Delta XYZ\text{ are }X\left(1,2\right),Y\left(3,6\right)\text{}\text{ and }Z\left(-1,4\right)$ .

Calculation:

Consider the coordinates of triangle $\Delta XYZ\text{ are }X\left(1,2\right),Y\left(3,6\right)\text{}\text{ and }Z\left(-1,4\right)$ .

The vertex matrix form is provided below,

  $\left[\begin{array}{ccc}1& 3& -1\\ 2& 6& 4\end{array}\right]$

According to the question the perimeter of the triangle is doubled.

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

So, if perimeter is to be doubled, each coordinate needs to be doubled.

For the new coordinates of the triangle the old vertex matrix must be multiplied with 2

Therefore,

  $2\left[\begin{array}{ccc}1& 3& -1\\ 2& 6& 4\end{array}\right]=\left[\begin{array}{ccc}2& 6& -2\\ 4& 12& 8\end{array}\right]$

Now, new coordinates are $X\text{'}\left(2,4\right),Y\text{'}\left(6,12\right)\text{}\text{ and }Z\text{'}\left(-2,8\right)$

Thus, the new coordinates of $\Delta X\text{'}Y\text{'}Z\text{'}\text{ are }X\text{'}\left(2,4\right),Y\text{'}\left(6,12\right)\text{}\text{ and }Z\text{'}\left(-2,8\right)$ .