Question

2015 > chapter 4: rotations > section exercises 4.3 > exercise 11
graph \\( \overline { x y } \\) with endpoints \\( x ( - 3,1 ) \\) and \\( y ( 4, - 5 ) \\) and its image after the composition.
translation: \\( ( x, y ) \
ightarrow ( x, y + 2 ) \\)
rotation: \\( 90 ^ { \circ } \\) about the origin
11

Explanation:

Step1: Apply the translation

For point \(X(-3,1)\):
Using the translation \((x,y)\to(x,y + 2)\), we substitute \(x=-3\) and \(y = 1\)
\(X'(-3,1+2)=X'(-3,3)\)
For point \(Y(4,-5)\):
Substitute \(x = 4\) and \(y=-5\) into \((x,y)\to(x,y + 2)\)
\(Y'(4,-5 + 2)=Y'(4,-3)\)

Step2: Apply the rotation

The rule for a \(90^{\circ}\) rotation about the origin is \((x,y)\to(-y,x)\)
For \(X'(-3,3)\):
Substitute \(x=-3\) and \(y = 3\) into \((x,y)\to(-y,x)\)
\(X''(-3,3)\to X''(-3,-3)\)
For \(Y'(4,-3)\):
Substitute \(x = 4\) and \(y=-3\) into \((x,y)\to(-y,x)\)
\(Y''(4,-3)\to Y''(3,4)\)

Answer:

The endpoints of the image of \(\overline{XY}\) after the composition (translation followed by rotation) are \(X''(-3,-3)\) and \(Y''(3,4)\)