Cumulative Assessment
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One angle and a side in the triangles are congruent. What else do you need to know?
Note that AB and DE are corresponding sides. How can you align their rotations given their original orientation?
See solution.
Composition of rigid motions:
&Rotation:90^(∘) about the origin
&Translation: (a,b)→ (a+13,b)
From the diagram, we know that ∠ B≅ ∠ E and BC≅ EF. If we can show that the second pair of sides that help create ∠ B and ∠ E are congruent as well, we can use the SAS Congruence Theorem to prove that the triangles are congruent;
Goal:
△ ABC≅ △ DEF
Since AB≅ DE, we can prove that △ ABC ≅ △ DEF by the SAS Congruence Theorem.
The triangles does not have the same orientation, so before we can translate △ ABC to △ DEF, we have to perform a rotation. From Part A, we know that AB and DE are corresponding sides. Additionally, since AB is vertical and DE is horizontal, we have to rotate △ ABC by 90^(∘) to give them the same orientation.
Rotating a figure by 90^(∘) counterclockwise about the origin, the coordinates of it's vertices change in the following way:
preimage (a,b) → image (- b,a)
| Point | (a,b) | (b,- a) |
|---|---|---|
| A | (2,8) | (8,- 2) |
| B | (2,5) | (5,- 2) |
| C | (5,6) | (- 6,5) |
Knowing the coordinates of the image vertices, we can graph it.
To map △ A'B'C' onto △ DEF, we have to translate the triangle by 13 units to the right. preimage (a,b) → image (a+13,b) Let's perform this translation.
