Question

examples 3 - 5
triangle abc has coordinates a(2, 0), b(-1, 5), and c(4, 3). determine the coordinates of the vertices of the image after each transformation.

1. reflection in x - axis
2. reflection in y - axis
3. translation along the vector (0, 2)
4. translation along the vector (3, - 4)
5. rotation 180° about the origin
6. rotation 90° counterclockwise about the origin

Explanation:

Step1: Reflection in x - axis

When reflecting a point \((x,y)\) in the \(x\) - axis, the rule is \((x,y)\to(x, - y)\).
For point \(A(2,0)\): \((2,0)\to(2,0)\)
For point \(B(-1,5)\): \((-1,5)\to(-1, - 5)\)
For point \(C(4,3)\): \((4,3)\to(4, - 3)\)

Step2: Reflection in y - axis

When reflecting a point \((x,y)\) in the \(y\) - axis, the rule is \((x,y)\to(-x,y)\).
For point \(A(2,0)\): \((2,0)\to(-2,0)\)
For point \(B(-1,5)\): \((-1,5)\to(1,5)\)
For point \(C(4,3)\): \((4,3)\to(-4,3)\)

Step3: Translation along the vector \(\langle0,2

angle\)
When translating a point \((x,y)\) along the vector \(\langle a,b
angle=(0,2)\), the rule is \((x,y)\to(x + a,y + b)=(x,y + 2)\).
For point \(A(2,0)\): \((2,0)\to(2,0 + 2)=(2,2)\)
For point \(B(-1,5)\): \((-1,5)\to(-1,5 + 2)=(-1,7)\)
For point \(C(4,3)\): \((4,3)\to(4,3 + 2)=(4,5)\)

Step4: Translation along the vector \(\langle3,-4

angle\)
When translating a point \((x,y)\) along the vector \(\langle a,b
angle=(3,-4)\), the rule is \((x,y)\to(x + a,y + b)=(x + 3,y-4)\).
For point \(A(2,0)\): \((2,0)\to(2+3,0 - 4)=(5,-4)\)
For point \(B(-1,5)\): \((-1,5)\to(-1 + 3,5-4)=(2,1)\)
For point \(C(4,3)\): \((4,3)\to(4 + 3,3-4)=(7,-1)\)

Step5: Rotation \(180^{\circ}\) about the origin

When rotating a point \((x,y)\) \(180^{\circ}\) about the origin, the rule is \((x,y)\to(-x,-y)\).
For point \(A(2,0)\): \((2,0)\to(-2,0)\)
For point \(B(-1,5)\): \((-1,5)\to(1,-5)\)
For point \(C(4,3)\): \((4,3)\to(-4,-3)\)

Step6: Rotation \(90^{\circ}\) counter - clockwise about the origin

When rotating a point \((x,y)\) \(90^{\circ}\) counter - clockwise about the origin, the rule is \((x,y)\to(-y,x)\).
For point \(A(2,0)\): \((2,0)\to(0,2)\)
For point \(B(-1,5)\): \((-1,5)\to(-5,-1)\)
For point \(C(4,3)\): \((4,3)\to(-3,4)\)

Answer:

1. \(A'(2,0)\), \(B'(-1,-5)\), \(C'(4,-3)\)
2. \(A'(-2,0)\), \(B'(1,5)\), \(C'(-4,3)\)
3. \(A'(2,2)\), \(B'(-1,7)\), \(C'(4,5)\)
4. \(A'(5,-4)\), \(B'(2,1)\), \(C'(7,-1)\)
5. \(A'(-2,0)\), \(B'(1,-5)\), \(C'(-4,-3)\)
6. \(A'(0,2)\), \(B'(-5,-1)\), \(C'(-3,4)\)