Question

1. given: ( ab cong bc ) if ( ab = 3x - 5 ) and ( bc = 2x + 8 ), find ( x ), ( ab ), ( bc ), ( ac ).

Explanation:

Step1: Set AB equal to BC (since \( AB \cong BC \))

Since \( AB \cong BC \), their lengths are equal. So we set up the equation:
\( 3x - 5 = 2x + 8 \)

Step2: Solve for x

Subtract \( 2x \) from both sides:
\( 3x - 2x - 5 = 2x - 2x + 8 \)
\( x - 5 = 8 \)

Then add 5 to both sides:
\( x - 5 + 5 = 8 + 5 \)
\( x = 13 \)

Step3: Find AB

Substitute \( x = 13 \) into the expression for \( AB \) (\( AB = 3x - 5 \)):
\( AB = 3(13) - 5 = 39 - 5 = 34 \)

Step4: Find BC

Since \( AB \cong BC \), \( BC = AB = 34 \) (or substitute \( x = 13 \) into \( BC = 2x + 8 \): \( BC = 2(13) + 8 = 26 + 8 = 34 \))

Step5: Find AC

From the diagram, \( AC = AB + BC \). So:
\( AC = 34 + 34 = 68 \)

Answer:

\( x = 13 \), \( AB = 34 \), \( BC = 34 \), \( AC = 68 \)