Question

which series of transformations can be used to justify this congruence statement?
rotate △tpc 90° counterclockwise about the origin and reflect the result across the x - axis.
rotate △tpc 180° about the origin and translate the result 2 units up.
reflect △tpc across the line y = x and reflect the result across the y - axis.
reflect △tpc across the y - axis and translate the result 2 units down.

Explanation:

Step1: Analyze each option

Let's consider the general rules of transformations for a point $(x,y)$:

• For a $90^{\circ}$ counter - clockwise rotation about the origin, the transformation is $(x,y)\to(-y,x)$. Then reflecting across the $x$ - axis changes $( - y,x)$ to $(-y,-x)$.
• For a $180^{\circ}$ rotation about the origin, the transformation is $(x,y)\to(-x,-y)$. Translating 2 units up changes $(-x,-y)$ to $(-x,-y + 2)$.
• Reflecting across the line $y=x$ changes $(x,y)$ to $(y,x)$. Reflecting across the $y$ - axis changes $(y,x)$ to $(-y,x)$.
• Reflecting across the $y$ - axis changes $(x,y)$ to $(-x,y)$. Translating 2 units down changes $(-x,y)$ to $(-x,y - 2)$.

Step2: Check the congruence

We need to check which sequence of transformations maps $\triangle TPC$ to the congruent triangle in the figure.
Let's assume the coordinates of the vertices of $\triangle TPC$ are $T(0,0)$, $P(4,4)$, $C(4,2)$.

• Option 1:
• For $P(4,4)$: After a $90^{\circ}$ counter - clockwise rotation about the origin, $P(4,4)\to(-4,4)$. After reflecting across the $x$ - axis, $(-4,4)\to(-4,-4)$.
• Option 2:
• For $P(4,4)$: After a $180^{\circ}$ rotation about the origin, $P(4,4)\to(-4,-4)$. After translating 2 units up, $(-4,-4)\to(-4,-2)$.
• Option 3:
• For $P(4,4)$: After reflecting across $y = x$, $P(4,4)\to(4,4)$. After reflecting across the $y$ - axis, $(4,4)\to(-4,4)$.
• Option 4:
• For $P(4,4)$: After reflecting across the $y$ - axis, $P(4,4)\to(-4,4)$. After translating 2 units down, $(-4,4)\to(-4,2)$.

By observing the figure and applying the transformations on the vertices of $\triangle TPC$, we find that rotating $\triangle TPC$ $180^{\circ}$ about the origin and translating the result 2 units up will map $\triangle TPC$ to the congruent triangle shown.

Answer:

Rotate $\triangle TPC$ $180^{\circ}$ about the origin and translate the result 2 units up.