Question

de bisects ac at point b. ac = 5y + 31 and ab = 6y + 12. find bc.

Explanation:

Step1: Use the bisection property

Since $\overline{DE}$ bisects $\overline{AC}$ at point $B$, we know that $AB = BC$ and $AC=AB + BC = 2AB$. So $5y + 31=2(6y + 12)$.

Step2: Solve the equation for y

Expand the right - hand side: $5y + 31 = 12y+24$.
Subtract $5y$ from both sides: $31=12y - 5y+24$, which simplifies to $31 = 7y+24$.
Subtract 24 from both sides: $7y=31 - 24=7$.
Divide both sides by 7: $y = 1$.

Step3: Find the length of AB

Substitute $y = 1$ into the expression for $AB$: $AB=6y + 12=6\times1+12=18$.

Step4: Find the length of BC

Since $BC = AB$, $BC = 18$.

Answer:

18