Question

13 write an algebraic formula for a rotation in which the preimage coordinates are a(1, - 3), b(1,1), c(5,1) and the image points are a(-1,3), b(-1,-1), c(-5,-1). 14. w(5,6) is the result of a 90° counter - clockwise rotation. find the coordinates of point w. 15. the points d(0,0), e(1,3), f(-4,5) represent the vertices of a polygon. find the coordinates of the image polygon after a rotation modeled by the rule (x,y)→(-y,x). 16. given the preimage coordinate d(7,2) and an image point d(2,-7) describe the counter - clockwise rotation and write an algebraic formula for the rotation that takes place on point d. 17. triangle abc is rotated 360° resulting in image points a(-5,5), b(3,1) and c(4,-4). what is the coordinate of preimage point b? 18. the line segment (overline{rs}) with endpoints r(9,3) and s(3,6) is rotated so the r is located at (-9,-3). write an algebraic rule that represents the counter - clockwise rotation and find s. 19. find the image point for m((\frac{1}{2},-5)) after a 270° counterclockwise rotation. 20. the points d(0,0), e(1,3), f(-4,5) represent the image vertices of a polygon. find the coordinates of the preimage polygon after a 270° counter - clockwise rotation. 21. given the algebraic rule (x,y)=(-x,-y) and the point e(7,-2), find e.

Explanation:

13

Step1: Analyze pre - image and image points

Pre - image points are \(A(1,-3)\), \(B(1,1)\), \(C(5,1)\) and image points are \(A'(-1,3)\), \(B'(-1,-1)\), \(C'(-5,-1)\). We can observe that \((x,y)\to(-x,-y)\). This is a \(180^{\circ}\) counter - clockwise rotation about the origin. The algebraic formula for a \(180^{\circ}\) counter - clockwise rotation about the origin is \((x,y)\to(-x,-y)\).

14

Step1: Recall the rule for \(90^{\circ}\) counter - clockwise rotation

The rule for a \(90^{\circ}\) counter - clockwise rotation about the origin is \((x,y)\to(-y,x)\). Let the coordinates of \(W\) be \((x,y)\). Then \((-y,x)=(5,6)\).

Step2: Solve for \(x\) and \(y\)

We have \(-y = 5\) and \(x = 6\), so \(y=-5\). The coordinates of \(W\) are \((6, - 5)\).

15

Step1: Apply the rotation rule \((x,y)\to(-y,x)\) to each vertex

For \(D(0,0)\): \((0,0)\to(0,0)\)
For \(E(1,3)\): \((1,3)\to(-3,1)\)
For \(F(-4,5)\): \((-4,5)\to(-5,-4)\)
The coordinates of the image polygon are \(D'(0,0)\), \(E'(-3,1)\), \(F'(-5,-4)\)

16

Answer:

Step1: Apply the rule \((x',y')=(-x,-y)\) to point \(E(7,-2)\)

Here \(x = 7\) and \(y=-2\)
\(x'=-7\) and \(y' = 2\)
So \(E'(-7,2)\)