Question

graph \\( \overline { a b } \\) with endpoints \\( a ( - 3,2 ) \\) and \\( b ( 3,8 ) \\) and its image after the composition of a rotation \\( 90 ^ { \circ } \\) about the origin and a translation along the vector \\( < 4,4 > \\).

Explanation:

Step1: Find the coordinates after rotation

The rule for a \(90^{\circ}\) rotation about the origin \((x,y)\to(-y,x)\).
For point \(A(-3,2)\), after rotation: \(A'(- 2,-3)\).
For point \(B(3,8)\), after rotation: \(B'(-8,3)\).

Step2: Find the coordinates after translation

The rule for translation along the vector \(\langle4,4
angle\) is \((x,y)\to(x + 4,y + 4)\).
For point \(A'(-2,-3)\), after translation: \(A''(-2+4,-3 + 4)=(2,1)\).
For point \(B'(-8,3)\), after translation: \(B''(-8 + 4,3+4)=(-4,7)\).

Answer:

The coordinates of the endpoints of the original segment are \(A(-3,2)\) and \(B(3,8)\). After a \(90^{\circ}\) rotation about the origin, the endpoints are \(A'(-2,-3)\) and \(B'(-8,3)\). After translation along the vector \(\langle4,4
angle\), the endpoints of the image segment are \(A''(2,1)\) and \(B''(-4,7)\).