a One angle and a side in the triangles are congruent. What else do you need to know?
B
b Note that AB and DE are corresponding sides. How can you align their rotations given their original orientation?
A
a See solution.
B
bComposition of rigid motions:
&Rotation:90^(∘) about the origin
&Translation: (a,b)→ (a+13,b)
Practice makes perfect
a 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
Note that AB is vertical and DE is horizontal. The length of a vertical side is the absolute value of the difference in the endpoints' y-coordinates. Similarly, the length of a horizontal side is determined as the absolute value of the difference in the endpoints' x-coordinates.
b 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.
Rotation
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)
Using this rule, we can determine the image vertices.
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.
Translation
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.
