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.

\\( x= \\)
\\( k l= \\)
\\( l m= \\)
\\( k m= \\)

Explanation:

Step1: Use the property of isosceles triangle

Since \(\angle K\cong\angle L\), then \(KM = LM\) (sides opposite to congruent angles in a triangle are congruent).
So, \(3x + 23=7x-37\).

Step2: Solve the equation for \(x\)

Subtract \(3x\) from both sides: \(23 = 7x-3x - 37\), which simplifies to \(23=4x - 37\).
Add \(37\) to both sides: \(23 + 37=4x\), so \(60 = 4x\).
Divide both sides by \(4\): \(x=\frac{60}{4}=15\).

Step3: Find the length of \(KL\)

Substitute \(x = 15\) into \(KL=9x - 40\). Then \(KL=9\times15-40=135 - 40=95\).

Step4: Find the length of \(LM\)

Substitute \(x = 15\) into \(LM = 7x-37\). Then \(LM=7\times15-37=105 - 37 = 68\).

Step5: Find the length of \(KM\)

Substitute \(x = 15\) into \(KM=3x + 23\). Then \(KM=3\times15+23=45 + 23=68\).

Answer:

\(x = 15\), \(KL = 95\), \(LM = 68\), \(KM = 68\)