Question

are triangles def and lnm similar if ln equals 4, mn equals 3, de equals 12, and fe equals 9?
yes by the sas similarity postulate
yes by the sss similarity postulate
no, the corresponding sides are not proportional
no, there is not enough info

Explanation:

Step1: Identify Right Angles

Triangles \( DEF \) and \( LNM \) are right - angled triangles (\( \angle E=\angle N = 90^{\circ}\)).

Step2: Check Proportionality of Sides

We are given \( LN = 4\), \( MN=3\), \( DE = 12\), \( FE = 9\).
Calculate the ratios of the corresponding sides:
\(\frac{LN}{FE}=\frac{4}{9}\) and \(\frac{MN}{DE}=\frac{3}{12}=\frac{1}{4}\)? Wait, no, let's re - check the correspondence. The right - angled sides: in \(\triangle DEF\), right - angled at \( E \), the legs are \( FE \) and \( DE \). In \(\triangle LNM\), right - angled at \( N \), the legs are \( MN \) and \( LN \).
So, \(\frac{FE}{LN}=\frac{9}{4}\) and \(\frac{DE}{MN}=\frac{12}{3} = 4\)? No, wait, the correct correspondence for SAS similarity (Side - Angle - Side) is that the ratio of two sides and the included angle (the right angle here) should be equal.
Let's find the ratios of the corresponding legs: \(\frac{FE}{MN}=\frac{9}{3}=3\) and \(\frac{DE}{LN}=\frac{12}{4} = 3\).
Since the ratio of the two pairs of corresponding legs is equal (\(\frac{FE}{MN}=\frac{DE}{LN}=3\)) and the included angles (\(\angle E\) and \(\angle N\)) are both right angles (equal), by the SAS (Side - Angle - Side) similarity postulate, the two triangles \( \triangle DEF\) and \( \triangle LNM\) are similar.

Answer:

YES, by the SAS Similarity Postulate