Question

in △klm, if ∠k ≅ ∠l, kl = 9x - 40, lm = 7x - 37, and km = 3x + 23, find the value of x and the measure of each side.

km =
lm =
kl =
x =
m∠s =
△xyz, ∠x ≅ ∠z, xy = 13 - 21, m

Explanation:

Step1: Set up the equation for congruent sides

Since \( \triangle KLM\) is isosceles (\(\angle K\cong\angle L\)), then \(KM = LM\).
We have \(KM = 3x + 23\) and \(LM=7x - 37\).
Set up the equation: \(3x + 23=7x - 37\).

Step2: Solve for \(x\)

Subtract \(3x\) from both sides: \(23 = 4x-37\).
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 \(KL = 9x-40\).
\(KL=9\times15 - 40=135 - 40 = 95\).

Step4: Find the length of \(LM\)

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

Step5: Find the length of \(KM\)

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

Answer:

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