Question

1. two triangles have side lengths 8 m, 12 m, and 16 m, and 4 m, 6 m, and 8 m, respectively. are these triangles similar? a. yes, by the aa criterion b. no, the angles do not match c. no, the sides are not proportional d. yes, by the sss criterion

Explanation:

Step1: Check side - length ratios

For two triangles with side - lengths \(a_1,b_1,c_1\) and \(a_2,b_2,c_2\), we check if \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\). Let the first triangle have side - lengths \(a_1 = 8\), \(b_1=12\), \(c_1 = 16\) and the second triangle have side - lengths \(a_2 = 4\), \(b_2 = 6\), \(c_2=8\). Calculate the ratios: \(\frac{8}{4}=\frac{12}{6}=\frac{16}{8}=2\).

Step2: Apply similarity criterion

Since the ratios of the corresponding side - lengths of the two triangles are equal, by the SSS (Side - Side - Side) similarity criterion, the two triangles are similar.

Answer:

d. Yes, by the SSS Criterion