Question

On a coordinate plane, 2 triangles are shown. Triangle ABC has points (-3, -1), (-1, 2), and (-5, 3). Triangle RST has points (1, 1), (3, 4), and (5, 0). Triangle RST is rotated 180° about the origin, and then translated up 3 units. Which congruency statement describes the figures?

1) ΔRST ≅ ΔACB
2) ΔRST ≅ ΔABC
3) ΔRST ≅ ΔBCA
4) ΔRST ≅ ΔBAC

Answer 1

After rotating Triangle RST by 180° and translating it up by 3 units, its coordinates match those of Triangle ABC when labeled as BAC. Hence, the correct congruency statement is Triangle RST ≃ Triangle BAC, which is answer choice (4).

Step-by-step explanation:

To determine which congruency statement correctly describes the relationship between Triangle ABC and the transformed Triangle RST, we must consider the effects of a 180° rotation around the origin followed by a translation up 3 units on Triangle RST. After rotating Triangle RST by 180°, the points will change signs (both x and y coordinates will be multiplied by -1), resulting in new coordinates: R becomes (-1, -1), S becomes (-3, -4), and T becomes (-5, 0). Following this rotation, translating the triangle up by 3 units will add 3 to the y-coordinates, making the new coordinates: R at (-1, 2), S at (-3, -1), and T at (-5, 3).

Now, comparing this transformed Triangle RST with Triangle ABC, which has points A at (-3, -1), B at (-1, 2), and C at (-5, 3), we see that the points now match exactly except the ordering of the letters. The transformed Triangle RST is R at (-1, 2), S at (-3, -1), and T at (-5, 3), which corresponds to points B, A, and C respectively of Triangle ABC. Therefore, it can be seen that Triangle RST is congruent to Triangle ABC when labeled as Triangle BAC.

The correct congruency statement is thus Triangle RST ≃ Triangle BAC, making the correct answer choice (4).