Question

find the length of the indicated side.

1. \\( \overline { d e } \\)
2. \\( \overline { k l } \\)

8.\\( \overline { d e } = \\)
9.\\( \overline { k l } = \\)

Explanation:

Step1: Solve for \(x\) in the first triangle

In \(\triangle DEF\), since the base angles are equal, the sides opposite them are equal. So \(DE = DF\).
Set \(5x - 12=3x - 4\).
Subtract \(3x\) from both sides: \(5x-3x - 12=3x - 3x-4\), which gives \(2x-12=-4\).
Add \(12\) to both sides: \(2x-12 + 12=-4 + 12\), so \(2x = 8\).
Divide both sides by \(2\): \(x=\frac{8}{2}=4\).

Step2: Find the length of \(DE\)

Substitute \(x = 4\) into the expression for \(DE\) (since \(DE=DF = 5x-12\) or \(DE\) can be calculated using the property of isosceles triangle).
\(DE=5\times4-12=20 - 12=8\).

Step3: Solve for \(x\) in the second triangle

In \(\triangle JKL\), since the base angles are equal, the sides opposite them are equal. So \(JK = JL\).
Set \(10x+3 = 3x+24\).
Subtract \(3x\) from both sides: \(10x-3x+3=3x - 3x+24\), which gives \(7x+3=24\).
Subtract \(3\) from both sides: \(7x+3 - 3=24 - 3\), so \(7x=21\).
Divide both sides by \(7\): \(x=\frac{21}{7}=3\).

Step4: Find the length of \(KL\)

Substitute \(x = 3\) into the expression for \(KL\) (\(KL = 3x+24\)).
\(KL=3\times3+24=9 + 24=33\).

Answer:

1. \(\overline{DE}=8\)
2. \(\overline{KL}=33\)