Chapter 4, Problem 10PT

To determine

To calculate: The coordinates of vertex $C\text{'}$ for the triangle $A\text{'}B\text{'}C\text{'}$ .

  $\begin{array}{cc}ABC& A\text{'}B\text{'}C\text{'}\\ A\left(3,4\right)& A\text{'}\left(5,3\right)\\ B\left(1,-2\right)& B\text{'}\left(3,-3\right)\\ C\left(5,-4\right)& C\text{'}\left(?,?\right)\end{array}$

The options are:

  $\begin{array}{l}A\left(-5,7\right)\\ B\left(7,-5\right)\\ C\left(4,-2\right)\\ D\left(7,-3\right)\end{array}$

Answer to Problem 10PT

The coordinates of vertex $C\text{'}$ are option $B\left(7,-5\right)$ .

Explanation of Solution

Given information:

The table that depicts the vertices of triangle

  $\begin{array}{cc}ABC& A\text{'}B\text{'}C\text{'}\\ A\left(3,4\right)& A\text{'}\left(5,3\right)\\ B\left(1,-2\right)& B\text{'}\left(3,-3\right)\\ C\left(5,-4\right)& C\text{'}\left(?,?\right)\end{array}$

Here the triangle $A\text{'}B\text{'}C\text{'}$ is the image of triangle $ABC$ .

Calculation:

Consider the table that depicts the vertices of triangle

  $\begin{array}{cc}ABC& A\text{'}B\text{'}C\text{'}\\ A\left(3,4\right)& A\text{'}\left(5,3\right)\\ B\left(1,-2\right)& B\text{'}\left(3,-3\right)\\ C\left(5,-4\right)& C\text{'}\left(?,?\right)\end{array}$

Here the triangle $A\text{'}B\text{'}C\text{'}$ is the image of triangle $ABC$ .

Observe that the x -coordinate is shifted 2 units right and y -coordinate is shifted 1 units down.

Therefore, $A\text{'}B\text{'}C\text{'}$ is the image of triangle $ABC$ that is shifted 2 units right and 1 unit down.

Coordinates of vertex are $C\left(5,-4\right)$ , so coordinates of vertex $C\text{'}$ are

  $C\text{'}\left(5+2,-4-1\right)=C\text{'}\left(7,-5\right)$

Thus, the coordinates of vertex $C\text{'}$ are option $B\left(7,-5\right)$ .