Question

using applying the converse of the side - splitter theorem is $overline{km} parallel overline{jn}$? why or why not? the triangle has vertices l, j, n. point k is on lj with lk = 16, kj = 10. point m is on ln with lm = 24, mn = 15. the options are: yes, because $\frac{16}{10}=\frac{15}{24}$. yes, because $\frac{16}{10}=\frac{24}{15}$. yes, because $\frac{10}{24}=\frac{15}{16}$. no, because $\frac{16}{10}
eq\frac{24}{15}$.

Explanation:

Step1: Recall Side - Splitter Theorem Converse

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

Step2: Identify the lengths

From the diagram, $LK = 16$, $KJ=10$, $LM = 24$, and $MN = 15$.

Step3: Calculate the ratios

Calculate $\frac{LK}{KJ}=\frac{16}{10}=\frac{8}{5}$ and $\frac{LM}{MN}=\frac{24}{15}=\frac{8}{5}$. 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 option with this statement)