Chapter 2.4, Problem 11E

To determine

To find: coordinates of the triangle by using scalar multiplication and then graph the pre-image and image on the same coordinate grid.

Answer to Problem 11E

  $\left(3,3\right),\left(3,12\right)\text{ and }\left(15,3\right)$

  

Explanation of Solution

Given information:

The vertices of triangle are $A\left(1,1\right),B\left(1,4\right)$ and $C\left(5,1\right)$ .

The scale factor is 3.

Calculation:

The rows will represent the x and y coordinates. The matrix for the given vertices $A,B$ and C are as follows:

  $\left[\begin{array}{ccc}\begin{array}{l}1\\ 1\end{array}& \begin{array}{l}1\\ 4\end{array}& \begin{array}{l}5\\ 1\end{array}\end{array}\right]$... (1)

Let the vertices of dilated triangle be represented by $A_1,B_1$ and $C_1$ .

Multiply the above matrix by 3 to get the vertices of the dilated triangle as follows:

  $\begin{array}{l}3\left[\begin{array}{ccc}\begin{array}{l}1\\ 1\end{array}& \begin{array}{l}1\\ 4\end{array}& \begin{array}{l}5\\ 1\end{array}\end{array}\right]=\left[\begin{array}{ccc}\begin{array}{l}1\left(3\right)\\ 1\left(3\right)\end{array}& \begin{array}{l}1\left(3\right)\\ 4\left(3\right)\end{array}& \begin{array}{l}5\left(3\right)\\ 1\left(3\right)\end{array}\end{array}\right]\\ =\left[\begin{array}{ccc}\begin{array}{l}3\\ 3\end{array}& \begin{array}{l}3\\ 12\end{array}& \begin{array}{l}15\\ 3\end{array}\end{array}\right]\end{array}$... (2)

The coordinates of the triangle by using scalar multiplication is $\left(3,3\right),\left(3,12\right)\text{ and }\left(15,3\right)$ .

The graph of the original (blue color) and the dilated (red color) triangles is as follows: