Answered

Triangle PQR has coordinates P(-2, 1), Q(-2, 7), and R(1, 4).
Triangle STU has coordinates S(-6, -2), T(6, -2), and U(0, -8).
Use coordinate geometry to determine if these two triangles are similar. If they are not similar, state the reason.
OA. The triangles are similar.
OB. The triangles are not similar because corresponding sides are not proportional.
OC. The triangles are not similar because they are congruent.
OD. The triangles are not similar because corresponding angles are not congruent.



Answer :

GinnyAnswer
To determine if the two triangles PQR and STU are similar, we have to check whether the sides of one triangle are proportional to the sides of the other. First, we calculate the lengths of the sides of each triangle using the distance formula: For triangle PQR: PQ = sqrt[(-2 - (-2))^2 + (7 - 1)^2] = sqrt[0 + 36] = sqrt[36] = 6 PR = sqrt[(-2 - 1)^2 + (1 - 4)^2] = sqrt[9 + 9] = sqrt[18] = 3√2 QR = sqrt[(-2 - 1)^2 + (7 - 4)^2] = sqrt[9 + 9] = sqrt[18] = 3√2 For triangle STU: ST = sqrt[(-6 - 6)^2 + (-2 - (-2))^2] = sqrt[(-12)^2 + 0] = sqrt[144] = 12 SU = sqrt[(-6 - 0)^2 + (-2 - (-8))^2] = sqrt[36 + 36] = sqrt[72] = 6√2 TU = sqrt[(6 - 0)^2 + (-2 - (-8))^2] = sqrt[36 + 36] = sqrt[72] = 6√2 Next, we compare the ratios of the corresponding sides: For triangle PQR: PQ/PR = 6 / (3√2) = 2√2 QR/PR = (3√2) / 3√2 = 1 For triangle STU: ST/SU = 12 / (6√2) = 2√2 TU/SU = 6√2 / 6√2 = 1 Comparing these ratios, we see that: - PQ/PR = ST/SU - QR/PR = TU/SU Thus, the ratios of the corresponding sides are equal, suggesting that the triangles are similar based on the Side-Side-Side (SSS) similarity postulate for triangles. Therefore, the correct answer is: OA. The triangles are similar.

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