Questions: For Exercises 1-5, use the diagram. 1. What composition of two rigid motions maps triangle ABC to triangle A'B'C'? For Exercises 2-5, find the coordinates of P' under each transformation. Suppose the equation of line m is y=2 and the equation of line n is x=-1. 2. T<-2,0> o rm 3. T<0,-5> o rn 4. T<0,2> o ry-axis 5. T<3,0> o rx-axis For Exercises 6-12, write a rigid motion that produces each image.

For Exercises 1-5, use the diagram.
1. What composition of two rigid motions maps triangle ABC to triangle A'B'C'?

For Exercises 2-5, find the coordinates of P' under each transformation. Suppose the equation of line m is y=2 and the equation of line n is x=-1.
2. T<-2,0> o rm
3. T<0,-5> o rn
4. T<0,2> o ry-axis
5. T<3,0> o rx-axis

For Exercises 6-12, write a rigid motion that produces each image.

Transcript text: For Exercises 1-5, use the diagram. 1. What composition of two rigid motions maps $\triangle A B C$ to $\triangle A^{\prime} B^{\prime} C^{\prime}$ ? For Exercises 2-5, find the coordinates of $P^{\prime}$ under each transformation. Suppose the equation of line $m$ is $y=2$ and the equation of line $n$ is $x=-1$. 2. $T_{\langle-2,0\rangle} \circ r_{m}$ 3. $T_{\langle 0,-5\rangle} \circ r_{n}$ 4. $T_{\langle 0,2\rangle} \circ r_{y \text {-axis }}$ 5. $T_{\langle 3,0\rangle} \circ r_{x \text {-axis }}$ For Exercises 6-12, write a rigid motion that produces each image.

Solution

Solution Steps

Step 1: Analyze the transformation from ΔABC to ΔA'B'C'

Triangle A'B'C' is obtained from triangle ABC by a rotation and a translation. First, reflect ΔABC across the x-axis. Then translate the reflected triangle 5 units to the left.

Step 2: Write the composition of rigid motions

The reflection across the x-axis is denoted by $r_{x-axis}$. The translation 5 units to the left is denoted by $T_{\langle -5, 0 \rangle}$. The composition of transformations is performed from right to left. So, the transformation is $T_{\langle -5, 0 \rangle} \circ r_{x-axis}$.

Step 3: Find the coordinates of P' under $T_{\langle-2,0\rangle} \circ r_{m}$

The coordinates of P are (1, 1). First, reflect P across line m (y=2). The y-coordinate of the reflected point is 2 + (2-1) = 3. The x-coordinate remains the same. So, the reflected point is (1, 3). Next, translate the point (1,3) by <-2, 0>. This means moving the point 2 units to the left. The new coordinates are (1 - 2, 3) = (-1, 3).

Step 4: Find the coordinates of P' under $T_{\langle 0,-5\rangle} \circ r_{n}$

The coordinates of P are (1, 1). First, reflect P across line n (x=-1). The x-coordinate of the reflected point is -1 - (1 - (-1)) = -1 - 2 = -3. The y-coordinate remains the same. So the reflected point is (-3, 1). Next, translate the point (-3, 1) by <0, -5>. This means moving the point 5 units down. The new coordinates are (-3, 1 - 5) = (-3, -4).