Question

On a coordinate plane, 2 triangles are shown. Triangle A B C has points (negative 1, negative 1), (2, negative 1), and (negative 1, negative 5). Triangle R S T has points (1, 1), (1, 5), and (4, 1).

Which best explains whether or not triangles RST and ACB are congruent?


a

The figures are congruent. ΔRST can be mapped to ΔACB by a reflection over the x-axis and a translation 2 units to the left.

b

The figures are congruent. ΔRST can be mapped to ΔACB by a reflection over the y-axis and a translation 2 units down.

c

The figures are not congruent. Point R corresponds to point A, but S corresponds to B and T corresponds to C.

d

The figures are not congruent. Point R does not correspond with point A.

Answer 1

a. The figures are congruent. ΔRST can be mapped to ΔACB by a reflection over the x-axis and a translation 2 units to the left.

Explanation:

You want an explanation of whether ∆RST ≅ ∆ACB and how congruence can be demonstrated.

LL Congruence

We note that both triangles are right triangles, and both have leg lengths of 3 and 4 units. This means the triangles are congruent by the LL congruence postulate (also known as SAS).

Rigid motions

The two triangles can be shown to be congruent if one can be mapped to the other by some combination of the rigid motions of translation, rotation, or reflection.

Relative to the right angle, the long leg points up in ∆RST and down in ∆ACB. Both short legs point to the right. This means one of the rigid motions required is reflection over a horizontal line, such as the x-axis.

Once ∆RST is reflected over the x-axis, we find point R' is 2 units to the right of point A. This indicates the mapping of ∆R'S'T' to ∆ACB will be complete after a translation of 2 units to the left.

ΔRST can be mapped to ΔACB by a reflection over the x-axis and a translation 2 units to the left.

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