Question

find ( x ) and then ( overline{ab} ) and ( overline{bc} ).
note: solving a quadratic equation will result in two possible answers. 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 as \(AC\) is marked as equal in length for both sides), we set \(x^{2}+5x=5x + 1\).
Subtract \(5x\) from both sides:
\(x^{2}+5x-5x=5x + 1-5x\)
\(x^{2}=1\)

Step2: Solve for \(x\)

Take the square root of both sides: \(x=\pm1\).
But when \(x = - 1\), \(AB=(-1)^{2}+5(-1)=1 - 5=-4\) (invalid as length can't be negative).
When \(x = 1\), \(AB=1^{2}+5\times1=6\) and \(BC=5\times1 + 1=6\)

Answer:

\(x = 1\)
\(\overline{AB}=6\)
\(\overline{BC}=6\)