Question

in the figure below, there are three right triangles. complete the following. (a) write a similarity statement relating the three right triangles. △ljm - △ - △ (b) complete each proportion. $\frac{jm}{jl}=\frac{lj}{}$ $\frac{kj}{kl}=\frac{}{km}$

Explanation:

Step1: Identify similar - triangles

In a right - triangle with an altitude drawn to the hypotenuse, the three right - triangles formed are similar. For right - triangle $\triangle LJM$ with altitude $KM$, we have $\triangle LJM\sim\triangle KJM\sim\triangle KJL$.

Step2: Use similarity to find proportions

For similar triangles, the ratios of corresponding sides are equal. In $\triangle LJM$ and $\triangle KJM$, if $\triangle LJM\sim\triangle KJM$, then $\frac{JM}{JL}=\frac{LJ}{KL}$ (corresponding sides of similar triangles). In $\triangle KJL$ and $\triangle KJM$, since they are similar, $\frac{KJ}{KL}=\frac{MJ}{KM}$ (corresponding sides of similar triangles).

Answer:

(a) $\triangle LJM\sim\triangle KJM\sim\triangle KJL$
(b) $\frac{JM}{JL}=\frac{LJ}{KL}$, $\frac{KJ}{KL}=\frac{MJ}{KM}$