Question

question 9 (multiple choice worth 1 points)
(03.05r lc)

are triangles def and lnm similar if ln equals 5, mn equals 3, de equals 10, and fe equals 6?

no, the corresponding sides are not proportional

Explanation:

Identify given values and corresponding parts

We are given two right triangles, \(\triangle DEF\) and \(\triangle LNM\), with right angles at \(\angle E\) and \(\angle N\).
The given side lengths are:

• \(DE = 10\)
• \(FE = 6\)
• \(LN = 5\)
• \(MN = 3\)

We want to determine if \(\triangle DEF \sim \triangle LNM\).

Check corresponding side ratios

Using the Proportional Sides concept, we compare the ratios of the corresponding legs forming the right angles:

$$ \frac{DE}{LN} = \frac{10}{5} = 2 $$

$$ \frac{FE}{MN} = \frac{6}{3} = 2 $$

Since the ratios are equal, the corresponding sides are proportional.

Apply similarity theorem

Using the SAS Similarity theorem:

• The included angles are congruent: \(\angle E \cong \angle N = 90^\circ\).
• The adjacent sides are proportional: \(\frac{DE}{LN} = \frac{FE}{MN} = 2\).

Therefore, the triangles are similar by SAS.

Answer:

• (A) Yes, by SAS Similarity (Correct answer)
• (B) No, the corresponding sides are not proportional
• (C) Yes, by AA Similarity
• (D) No, the corresponding angles are not congruent