Question

determine if the two figures below are congruent. use the drop - down menus below to justify why. which point on figure 2 corresponds to point ( r ) on figure 1?

Explanation:

Step1: Find the coordinates of points in Figure 1

Let's assume the coordinates of \(T(-5,4)\), \(Q(-3,9)\), \(S(-2,7)\), \(R(-1,2)\) in Figure 1.

Step2: Find the coordinates of points in Figure 2

Assume the coordinates of \(E(6,7)\), \(D(8,9)\), \(G(7,2)\), \(F(10,5)\) in Figure 2.

Step3: Check the distance between corresponding points

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For example, distance \(TQ=\sqrt{(-3 + 5)^2+(9 - 4)^2}=\sqrt{4 + 25}=\sqrt{29}\)
Distance \(ED=\sqrt{(8 - 6)^2+(9 - 7)^2}=\sqrt{4 + 4}=\sqrt{8}\) (This is wrong approach, actually we can check the transformation.
We can observe that Figure 1 is translated 7 units to the right.
For point \(R(-1,2)\), after translation \(x=-1+7 = 6\), \(y = 2\) (no vertical translation in this case). But wait, another way:
We can check the shape and side - length. Since the two figures have the same shape and size (by counting the grid - based side - lengths, for example, the vertical and horizontal distances between adjacent points in the two figures are the same after considering translation).
If we consider the transformation (translation), for point \(R\) in Figure 1, when we move Figure 1 7 units to the right (since \(x\) - coordinate of \(R\) is \(-1\) and \(x\) - coordinate of \(G\) is \(7\), \(-1+8=7\) (wait, no, better:
We can check the relative position of the points. The two figures are congruent (same shape and size).
For the correspondence:
We can see that \(R\) corresponds to \(G\) because of the translation (movement of the entire figure).

Answer:

The two figures are congruent. Point \(R\) in Figure 1 corresponds to point \(G\) in Figure 2.