a What does the angle at ∠A look like? What would this mean for the slopes of AB and AC?
B
b What can you say about B'C and BC?
A
aDiagram:
Type: â–³ ABC is a right triangle.
B
b An isosceles triangle, see solution.
Practice makes perfect
a Let's draw â–³ ABC using the given information.
Examining the diagram, it would appear that △ ABC is a right triangle with a right angle at ∠A. To investigate this we must check if AB and AC are perpendicular. If they are, the slope of AB and AC should multiply to - 1.
m_(AB)m_(AC)? =- 1
Let's determine the slopes of AB and AC by substituting the coordinates of the segment's endpoints into the Slope Formula.
As we can see, AB and AC are perpendicular which means ∠A is a right angle. Therefore, △ ABC must be a right triangle.
b Since we are reflecting the triangle in AC, only B will be affected. From Part A, we know that AB⊥ AC. Therefore, to reflect B across AC we have to extend AB across AC until AB=AB'.
Notice that △ ABC and △ AB'C share AC as a side. Therefore, these must be congruent according to the Reflexive Property of Congruence. Now we have enough information to prove that △ ABC≅ △ AB'C by the SAS (Side-Angle-Side) Congruence Theorem. Since BC and B'C are corresponding sides, they must be congruent.
