Question

congruent triangles pqr and pqr are shown on the coordinate plane.
which series of transformations maps triangle pqr onto triangle pqr?
a a rotation 90° counterclockwise about the origin and a translation up 4 units
b a reflection over the y - axis and a translation up 4 units
c a reflection over the x - axis and a rotation 90° clockwise about the origin
d a translation right 7 units and a reflection over the x - axis

Explanation:

Step1: Analyze the reflection

First, consider the reflection. Reflecting triangle \(PQR\) over the \(y -\)axis changes the \(x -\)coordinates of its vertices. For example, if a point \((x,y)\) is reflected over the \(y -\)axis, it becomes \((-x,y)\).

Step2: Analyze the translation

After reflecting over the \(y -\)axis, we need to translate the triangle. A translation up \(4\) units changes the \(y -\)coordinate of each vertex. If a point \((x,y)\) is translated up \(4\) units, it becomes \((x,y + 4)\).

Let's check the vertices:

• Suppose \(P(-6,-7)\), after reflection over the \(y -\)axis: \((6,-7)\), then after translation up \(4\) units: \((6,- 3)\)
• Suppose \(Q(-7,-2)\), after reflection over the \(y -\)axis: \((7,-2)\), then after translation up \(4\) units: \((7,2)\)
• Suppose \(R(-1,-2)\), after reflection over the \(y -\)axis: \((1,-2)\), then after translation up \(4\) units: \((1,2)\)

For option A: Rotation \(90^{\circ}\) counter - clockwise about the origin: \((x,y)\to(-y,x)\), which does not give the correct mapping.
For option C: Reflection over the \(x -\)axis \((x,y)\to(x,-y)\) and then rotation \(90^{\circ}\) clockwise \((x,y)\to(y,-x)\) does not give the correct mapping.
For option D: Translation right \(7\) units \((x,y)\to(x + 7,y)\) and reflection over the \(x -\)axis \((x,y)\to(x,-y)\) does not give the correct mapping.

Answer:

B. a reflection over the \(y -\)axis and a translation up \(4\) units