Question

find ( x ) and then ( overline{ab} ) and ( overline{bc} ).
note: always make sure the value of ( x ) produces positive lengths.
( x=)
( overline{ab}=)
( overline{bc}=)

Explanation:

Step1: Set up the equation

Since \(AB = BC\) (the triangle is isosceles), we have \(3x + 5=4x + 2\).

Step2: Solve for \(x\)

Subtract \(3x\) from both sides: \(3x+5 - 3x=4x + 2-3x\), which gives \(5=x + 2\).
Then subtract \(2\) from both sides: \(5-2=x+2 - 2\), so \(x = 3\).

Step3: Find \(AB\)

Substitute \(x = 3\) into \(AB=3x + 5\). Then \(AB=3\times3+5=9 + 5=14\).

Step4: Find \(BC\)

Substitute \(x = 3\) into \(BC=4x + 2\). Then \(BC=4\times3+2=12 + 2=14\).

Answer:

\(x = 3\), \(\overline{AB}=14\), \(\overline{BC}=14\)