Question

1. in \\( \triangle k l m \\), if \\( \angle k \cong \angle l, k l = 9 x - 40, l m = 7 x - 37 \\), and \\( k m = 3 x + 23 \\), find the value of \\( x \\) and the measure of each side.
2. in \\( \triangle x y z \\), if \\( \angle x \cong \angle z, x y = 13 x - 21, y z = 8 x - 6 \\), and \\( x z = x + 4 \\), find the value of \\( x \\) and the measure of each side.

Explanation:

Step1: <Set up equation for isosceles triangle>

Since \(\angle K\cong\angle L\) in \(\triangle KLM\), \(KL = KM\).
So, \(9x - 40=3x + 23\).

Step2: <Solve the equation for \(x\)>

Subtract \(3x\) from both sides: \(9x-3x - 40=3x-3x + 23\), which gives \(6x-40 = 23\).
Add \(40\) to both sides: \(6x-40 + 40=23 + 40\), so \(6x=63\).
Divide both sides by \(6\): \(x=\frac{63}{6}=\frac{21}{2}=10.5\).

Step3: <Find the lengths of sides>

For \(KL\): Substitute \(x = 10.5\) into \(KL=9x - 40\), \(KL=9\times10.5-40=94.5 - 40 = 54.5\).
For \(LM\): Substitute \(x = 10.5\) into \(LM = 7x-37\), \(LM=7\times10.5-37=73.5 - 37 = 36.5\).
For \(KM\): Substitute \(x = 10.5\) into \(KM=3x + 23\), \(KM=3\times10.5+23=31.5+23 = 54.5\).

Answer:

\(x = 10.5\), \(KL = 54.5\), \(LM = 36.5\), \(KM = 54.5\)