Question

is $overline{km} parallel overline{jn}$? why or why not?

$circ$ no, because $\frac{16}{10}
eq \frac{24}{15}$.

$circ$ yes, because $\frac{10}{24} = \frac{15}{16}$.

$circ$ yes, because $\frac{16}{10} = \frac{15}{24}$.

$circ$ yes, because $\frac{16}{10} = \frac{24}{15}$. (diagram: triangle with points j, k, l, m, n; jk=10, kl=16, lm=24, mn=15, k and m connected)

Explanation:

To determine if \(\overline{KM} \parallel \overline{JN}\), we use the Basic Proportionality Theorem (Thales' theorem), which states that if a line divides two sides of a triangle proportionally, then it is parallel to the third side.

Step 1: Identify the segments

In \(\triangle LJN\), the line \(KM\) intersects \(LJ\) at \(K\) and \(LN\) at \(M\). The lengths are:

• \(LK = 16\), \(KJ = 10\)
• \(LM = 24\), \(MN = 15\)

Step 2: Check the ratios

We need to check if \(\frac{LK}{KJ}=\frac{LM}{MN}\)

Calculate \(\frac{LK}{KJ}\):
\(\frac{LK}{KJ}=\frac{16}{10}=\frac{8}{5}\)

Calculate \(\frac{LM}{MN}\):
\(\frac{LM}{MN}=\frac{24}{15}=\frac{8}{5}\)

Since \(\frac{16}{10}=\frac{24}{15}\) (both simplify to \(\frac{8}{5}\)), by the Basic Proportionality Theorem, \(\overline{KM} \parallel \overline{JN}\)

Answer:

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