Question

select the sequence of transformations that will carry triangle a onto triangle a
a reflect over x - axis, translate 2 units up, then 10 units left
b reflect over x - axis, reflect over y - axis, then rotate 90° clockwise
c reflect over y - axis, reflect over x - axis, then rotate 180° clockwise
d reflect over x - axis, reflect over y - axis, then translate 2 units up

Explanation:

Step1: Analyze option A

• Reflect over \(x -\)axis: flips the triangle vertically.
• Translate 2 units up: moves the triangle 2 units in the positive \(y -\)direction.
• Translate 10 units left: moves the triangle 10 units in the negative \(x -\)direction.

Let's assume a vertex of triangle \(A\) is \((x,y)\). After reflection over \(x -\)axis, it becomes \((x,-y)\). After translating 2 units up, it is \((x,-y + 2)\). After translating 10 units left, it is \((x-10,-y + 2)\).

Step2: Analyze option B

• Reflect over \(x -\)axis: \((x,y)\to(x,-y)\).
• Reflect over \(y -\)axis: \((x,-y)\to(-x,-y)\).
• Rotate \(90^{\circ}\) clockwise: Using the rotation formula \((x,y)\to(y,-x)\), for \((-x,-y)\) it becomes \((-y,x)\).

Step3: Analyze option C

• Reflect over \(y -\)axis: \((x,y)\to(-x,y)\).
• Reflect over \(x -\)axis: \((-x,y)\to(-x,-y)\).
• Rotate \(180^{\circ}\) clockwise: Using the rotation formula \((x,y)\to(-x,-y)\), for \((-x,-y)\) it becomes \((x,y)\) (not correct).

Step4: Analyze option D

• Reflect over \(x -\)axis: \((x,y)\to(x,-y)\).
• Reflect over \(y -\)axis: \((x,-y)\to(-x,-y)\).
• Translate 2 units up: \((-x,-y)\to(-x,-y + 2)\).

Let's take a sample vertex of triangle \(A\) say \((4,2)\)

• Option A:
• Reflect over \(x -\)axis: \((4,-2)\)
• Translate 2 units up: \((4,0)\)
• Translate 10 units left: \((- 6,0)\)
• Option B:
• Reflect over \(x -\)axis: \((4,-2)\)
• Reflect over \(y -\)axis: \((-4,-2)\)
• Rotate \(90^{\circ}\) clockwise: \((-2,4)\)
• Option D:
• Reflect over \(x -\)axis: \((4,-2)\)
• Reflect over \(y -\)axis: \((-4,-2)\)
• Translate 2 units up: \((-4,0)\)

If we assume the coordinates of triangle \(A\) vertices and check the transformations, for option A:
Suppose triangle \(A\) has vertices \((4,2)\), \((6,2)\), \((6,4)\)

• Reflect over \(x -\)axis: \((4,-2)\), \((6,-2)\), \((6,-4)\)
• Translate 2 units up: \((4,0)\), \((6,0)\), \((6,-2)\)
• Translate 10 units left: \((-6,0)\), \((-4,0)\), \((-4,-2)\) which matches the position of \(A'\)

Answer:

A. reflect over x - axis, translate 2 units up, then 10 units left