Question

in the figure below, there are three right trangles. complete the following. (a) write a similiarity statement relating the three right triangles. (\triangle mklsim\triangle jmlsim\triangle jkl) (b) complete each proportion. (\frac{mj}{lj}=\frac{lj}{square}) (\frac{kj}{kl}=\frac{square}{mk})

Explanation:

Step1: Use the similarity of triangles

Since \(\triangle MKL\sim\triangle JML\sim\triangle JKL\), for \(\frac{MJ}{LJ}=\frac{LJ}{?}\), using the property of similar right - angled triangles (geometric mean theorem in right - angled triangles). In \(\triangle JML\) and \(\triangle JKL\), \(\angle MJL=\angle LJK\) (common angle) and \(\angle JML = \angle JKL=90^{\circ}\). By the geometric mean theorem, in a right - angled triangle, the altitude to the hypotenuse is the geometric mean between the segments of the hypotenuse. So \(\frac{MJ}{LJ}=\frac{LJ}{KJ}\).

Step2: For the second proportion \(\frac{KJ}{KL}=\frac{?}{MK}\)

Since \(\triangle JKL\sim\triangle MKL\), \(\angle KJL=\angle MKL\) (because \(\angle JKL = \angle KML = 90^{\circ}\) and \(\angle L\) is common to \(\triangle JKL\) and \(\triangle MKL\)). By the similarity of \(\triangle JKL\) and \(\triangle MKL\), the ratios of corresponding sides are equal. So \(\frac{KJ}{KL}=\frac{JL}{MK}\)

Answer:

For \(\frac{MJ}{LJ}=\frac{LJ}{KJ}\), the blank is \(KJ\). For \(\frac{KJ}{KL}=\frac{JL}{MK}\), the blank is \(JL\)