Question

find the value of a in the two similar triangles.
triangle lkm: right - angled at l, side lk = 120, side lm = a, side km = 150.
triangle onp: right - angled at o, side on = 80, side op = 60, side np = 100.

Explanation:

Step1: Identify Similar Triangles

Since the triangles are similar, their corresponding sides are proportional. The right triangles \( \triangle KLM \) and \( \triangle NOP \) have right angles at \( L \) and \( O \) respectively. The sides \( KL = 120 \), \( LM = a \), \( KM = 150 \) in \( \triangle KLM \); and \( NO = 80 \), \( OP = 60 \), \( NP = 100 \) in \( \triangle NOP \).

Step2: Set Up Proportion

We can set up the proportion using the corresponding sides. Let's use the legs and the hypotenuse or the legs. Let's take the legs \( KL \) and \( NO \), and the legs \( LM \) and \( OP \). So, \( \frac{KL}{NO}=\frac{LM}{OP} \).

Substituting the known values: \( \frac{120}{80}=\frac{a}{60} \).

Step3: Solve for \( a \)

First, simplify \( \frac{120}{80}=\frac{3}{2} \). Then, we have \( \frac{3}{2}=\frac{a}{60} \). Cross - multiply: \( 2a = 3\times60 \). So, \( 2a = 180 \). Divide both sides by 2: \( a=\frac{180}{2}=90 \).

We can also check with the hypotenuses. \( \frac{KM}{NP}=\frac{150}{100}=\frac{3}{2} \), and \( \frac{KL}{NO}=\frac{120}{80}=\frac{3}{2} \), so the proportion holds.

Answer:

\( a = 90 \)