2.05 transformations and congruence activity on FLVS

The steps given in the question are;

Step 1: The type of triangle chosen is the equilateral equiangular triangle with rotation to prove SAS congruence

Step 2

The chosen type of triangle, the equilateral, equiangular triangle, ΔABC, is constructed on MS Excel using the points A(0, 0), B(6, 0), and C(3, 3·√3)

The lengths of the sides of the triangle are verified as follows;

AB = 6 - 0 = 6

AC = √(3² + (3·√3)²) = 6

CB = √((6 - 3)² + (3·√3)²) = √(3² + (3·√3)²) = 6

Therefore, ΔABC is an equilateral triangle

An equilateral triangle = An equiangular triangle

∴ ∠A  = ∠B = ∠C = 60°

The coordinates of the image of the preimage (x, y), following a rotation transformation of 180° clockwise is given as follows;

(x, y) 180°CW → (-x, -y)

Therefore, the coordinates of the image ΔA'B'C' of the preimage triangle ΔABC after a 180° rotation transformation are;

A'(0, 0), B'(-6, 0), and C'(-3, -3·√3)

The lengths of the sides of the triangle ΔA'B'C' are;

A'B' = 0 - (-6) = 6

A'C' = √((-3)² + (-3·√3)²) = 6

C'B' = √((-6 - (-3))² + (-3·√3)²) = √(3² + (3·√3)²) = 6

Therefore, ΔA'B'C' is an equilateral triangle and therefore an equiangular triangle

∴ ∠A' = ∠B' = ∠C' = 60°

Therefore, the lengths of the sides of the triangle ΔABC are equal to the lengths of the sides of the triangle ΔA'B'C', and we have;

AB = A'B' therefore, AB ≅ A'B'

AC = A'C', therefore, AC ≅ A'C'

∠A = ∠A' = 60°, therefore, ∠A ≅ ∠A'

Therefore, triangle ΔABC is congruent to ΔA'B'C', by Side-Angle-Side, also known as, SAS rule of congruency

Learn more about rigid transformations here;