Question

what jam cant you eat?
write the letter of each answer in the box containing
the exercise number.
the vertices of a triangle are a(2, 3), b(7, 4), and
c(6, 1). rotate the triangle as described. find the
coordinates of the image.

1. 90° clockwise about the origin
2. 180° about the origin
3. 90° counterclockwise about the origin
4. 180° about vertex a

Explanation:

Step1: Rotation rules

• For a \(90^{\circ}\) clockwise rotation about the origin \((x,y)\to(y, -x)\)
• For a \(180^{\circ}\) rotation about the origin \((x,y)\to(-x,-y)\)
• For a \(90^{\circ}\) counter - clockwise rotation about the origin \((x,y)\to(-y,x)\)
• For a \(180^{\circ}\) rotation about a point \((a,b)\): First translate the point \((x - a,y - b)\), then rotate \((-(x - a),-(y - b))\), and then translate back \((a-(x - a),b-(y - b))=(2a - x,2b - y)\)

Step2: \(90^{\circ}\) clockwise rotation about the origin

• For \(A(2,3)\): \(x = 2,y = 3\), new coordinates \((3,-2)\)
• For \(B(7,4)\): \(x = 7,y = 4\), new coordinates \((4,-7)\)
• For \(C(6,1)\): \(x = 6,y = 1\), new coordinates \((1,-6)\)

Step3: \(180^{\circ}\) rotation about the origin

• For \(A(2,3)\): \(x = 2,y = 3\), new coordinates \((-2,-3)\)
• For \(B(7,4)\): \(x = 7,y = 4\), new coordinates \((-7,-4)\)
• For \(C(6,1)\): \(x = 6,y = 1\), new coordinates \((-6,-1)\)

Step4: \(90^{\circ}\) counter - clockwise rotation about the origin

• For \(A(2,3)\): \(x = 2,y = 3\), new coordinates \((-3,2)\)
• For \(B(7,4)\): \(x = 7,y = 4\), new coordinates \((-4,7)\)
• For \(C(6,1)\): \(x = 6,y = 1\), new coordinates \((-1,6)\)

Step5: \(180^{\circ}\) rotation about vertex \(A(2,3)\)

• For \(A(2,3)\): It remains \((2,3)\)
• For \(B(7,4)\): \(x = 7,y = 4\), new coordinates \((2\times2 - 7,2\times3 - 4)=(-3,2)\)
• For \(C(6,1)\): \(x = 6,y = 1\), new coordinates \((2\times2 - 6,2\times3 - 1)=(-2,5)\)

Answer:

1. \(A'(3,-2)\), \(B'(4,-7)\), \(C'(1,-6)\)
2. \(A'(-2,-3)\), \(B'(-7,-4)\), \(C'(-6,-1)\)
3. \(A'(-3,2)\), \(B'(-4,7)\), \(C'(-1,6)\)
4. \(A'(2,3)\), \(B'(-3,2)\), \(C'(-2,5)\)