Question

1. in the problem below, \\( \triangle a b c \cong \triangle a d c \\). solve for \\( x \\). then, use that value to find the length of \\( b c \\). (1 point)

Explanation:

Step1: Use the property of congruent triangles

Since \(\triangle ABC\cong\triangle ADC\), their corresponding sides are equal. So \(BC = DC\).

Step2: Set up the equation

We have \(3x + 30=12x - 15\).

Step3: Solve the equation for \(x\)

Subtract \(3x\) from both sides: \(30 = 9x-15\).
Add \(15\) to both sides: \(45 = 9x\).
Divide both sides by \(9\): \(x = 5\).

Step4: Find the length of \(BC\)

Substitute \(x = 5\) into the expression for \(BC\) (\(12x - 15\)).
\(BC=12\times5 - 15=60 - 15 = 45\).

Answer:

\(x = 5\) and the length of \(BC\) is \(45\).