Question

,..........An illustration of a coordinate plane with four quadrants shows the x and y axes extending from negative 10 to 10 in increments of 1. Two triangles, upper A upper B upper C and upper E upper G upper H, are plotted on the graph. The triangle upper A upper B upper C has its vertices marked with closed points at upper A left parenthesis 1 comma negative 3 right parenthesis, upper B left parenthesis 1 comma negative 1 right parenthesis, and upper C left parenthesis 4 comma negative 3 right parenthesis. The triangle upper E upper G upper H has its vertices marked with closed points at upper E left parenthesis negative 2 comma 5 right parenthesis, upper G left parenthesis negative 2 comma 2 right parenthesis, and upper H left parenthesis negative 4 comma 2 right parenthesis. In triangle upper E upper G upper H, side upper E upper G is labeled as h, side upper G upper H is labeled as e, and side upper H upper E is labeled as g.

△ABC
is congruent to △GHE
. Describe a sequence of transformations from △ABC
to △GHE
that proves their congruence.

Answer 1

One convenient sequence:

1. Rotate triangle ABC 90° counterclockwise about A(1,-3). This sends
- A to A (1,-3),
- B(1,-1) to B' = (-1,-3),
- C(4,-3) to C' = (1,0).

2. Translate the rotated image 3 units left and 5 units up (vector = ⟨-3,5⟩). This sends
- A → A + ⟨-3,5⟩ = (-2,2) = G,
- B' → (-1,-3) + ⟨-3,5⟩ = (-4,2) = H,
- C' → (1,0) + ⟨-3,5⟩ = (-2,5) = E.

Thus A→G, B→H, C→E and △ABC is carried exactly onto △GHE, proving they are congruent.