Question

practice applying the side-splitter theorem, its converse, and the triangle midsegment theorem. is \\(\overline{km} \parallel \overline{jn}\\)? why or why not? \\(\bigcirc\\) no, because \\(\frac{16}{10} \
eq \frac{24}{15}\\). \\(\bigcirc\\) yes, because \\(\frac{10}{24} = \frac{15}{16}\\). \\(\bigcirc\\) yes, because \\(\frac{16}{10} = \frac{15}{24}\\) \\(\bigcirc\\) yes, because \\(\frac{16}{10} = \frac{24}{15}\\).

Explanation:

Step1: Recall the Converse of the Side - Splitter Theorem

The converse of the side - splitter theorem states that if a line divides two sides of a triangle proportionally, then the line is parallel to the third side. For triangle \( \triangle LJN \), with a line \( KM \) intersecting \( LJ \) at \( K \) and \( LN \) at \( M \), we need to check if \( \frac{LK}{KJ}=\frac{LM}{MN} \).

We know that \( LK = 16 \), \( KJ=10 \), \( LM = 24 \), and \( MN = 15 \).

Step2: Check the Proportions

Calculate the ratios:

• The ratio of \( LK \) to \( KJ \) is \( \frac{LK}{KJ}=\frac{16}{10} \)
• The ratio of \( LM \) to \( MN \) is \( \frac{LM}{MN}=\frac{24}{15} \)

Simplify both ratios:

• \( \frac{16}{10}=\frac{8}{5} = 1.6 \)
• \( \frac{24}{15}=\frac{8}{5}=1.6 \)

Since \( \frac{16}{10}=\frac{24}{15} \), by the converse of the side - splitter theorem, \( KM\parallel JN \).

Answer:

Yes, because \(\frac{16}{10}=\frac{24}{15}\) (the last option)