Question

The vertices of ΔMNO are M (1, 3), N (4, 9), and O (7, 3). The vertices of ΔPQR are P (3, 0), Q (4, 2), and R (5, 0). Which conclusion is true about the triangles?

The ratio of their corresponding sides is 1:2.
The ratio of their corresponding angles is 1:2.
They are similar by the definition of similarity in terms of a dilation.
They are congruent by the definition of congruence in terms of rigid motions.

Answer 1

To determine if ΔMNO and ΔPQR are similar or congruent, analyze the corresponding angles and side lengths. Dilation suggests similarity, while identical measures via rigid motions infer congruence. Calculations of sides and angles would be necessary to confirm the correct conclusion.

Step-by-step explanation:

The question involves comparing the similarities or congruences of two triangles, ΔMNO and ΔPQR, based on their vertices. To determine if they are similar or congruent, one must analyze the corresponding angles and side lengths of each triangle. If two triangles are similar, their corresponding angles are equal and the sides are proportional. In contrast, congruence implies that both the angles and sides are identical in measure when superimposed via rigid motions (like rotation, reflection, or translation).

To evaluate the corresponding sides and angles, one can use the distance formula to find the lengths of the sides of both triangles and the slope of these sides to determine if the angles are equal. However, a quick observation shows that ΔMNO appears to be a stretched version of ΔPQR, which suggests a dilation has occurred. Dilation is a transformation that produces a figure similar to the original by scaling it uniformly.

Upon computing and comparing the sides and angles, one will be able to identify whether ΔMNO and ΔPQR are similar by dilation or congruent by rigid motions. However, without these specific calculations, we cannot confidently confirm one of the conclusions presented in the question.