Question 6 (Multiple Choice Worth 3 Points) (02.07 MC) Triangle ABC maps to A'B'C' as shown. A(-5;-3), B(-2;-3) C(-2;-8) A'(3;-5) B'(3;-2) C'(8;-2) Is triangle ABC congruent to triangle A'B'C'? Why, or why not?
Answer (2)
To determine whether triangle A BC is congruent to triangle A ′ B ′ C ′ , we need to check if there's a congruence transformation (such as a translation, rotation, reflection, or a combination of these) that maps one triangle onto the other while preserving the side lengths and angles.
First, let's calculate the side lengths of both triangles using the distance formula:
The distance between two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by: d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2
For triangle A BC :
A B :
d A B = (( − 2 ) − ( − 5 ) ) 2 + (( − 3 ) − ( − 3 ) ) 2 = 3 2 + 0 2 = 3
BC :
d BC = (( − 2 ) − ( − 2 ) ) 2 + (( − 8 ) − ( − 3 ) ) 2 = 0 + ( − 5 ) 2 = 5
C A :
d C A = (( − 5 ) − ( − 2 ) ) 2 + (( − 3 ) − ( − 8 ) ) 2 = 3 2 + ( − 5 ) 2 = 9 + 25 = 34
For triangle A ′ B ′ C ′ :
A ′ B ′ :
d A ′ B ′ = (( 3 ) − ( 3 ) ) 2 + (( − 2 ) − ( − 5 ) ) 2 = 0 + 3 2 = 3
B ′ C ′ :
d B ′ C ′ = (( 8 ) − ( 3 ) ) 2 + (( − 2 ) − ( − 2 ) ) 2 = 5 2 + 0 = 5
C ′ A ′ :
d C ′ A ′ = (( 8 ) − ( 3 ) ) 2 + (( − 2 ) − ( − 5 ) ) 2 = 5 2 + 3 2 = 34
Since the side lengths of triangle A BC and triangle A ′ B ′ C ′ are the same, they are congruent by the Side-Side-Side (SSS) Congruence Postulate.
Thus, the correct choice is: Yes, triangle A BC is congruent to triangle A ′ B ′ C ′ because their corresponding side lengths are equal.
Triangle ABC is congruent to triangle A'B'C' because all corresponding side lengths are equal. The lengths of the sides were calculated and found to be 3, 5, and \sqrt{34} for both triangles. Thus, the triangles are congruent by the SSS Congruence Postulate.
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