Question

1. if (overline{bd}congoverline{bc}), (bd = 5x - 26), (bc = 2x+1), and (ac = 43), find (ab).

Explanation:

Step1: Set up the equation from equal - length segments

Since $\overline{BD}=\overline{BC}$, we have $5x - 26=2x + 1$.

Step2: Solve the equation for x

Subtract $2x$ from both sides: $5x-2x-26=2x - 2x+1$, which simplifies to $3x-26 = 1$. Then add 26 to both sides: $3x-26 + 26=1 + 26$, so $3x=27$. Divide both sides by 3: $x=\frac{27}{3}=9$.

Step3: Find the length of BC

Substitute $x = 9$ into the expression for $BC$. $BC=2x + 1=2\times9+1=18 + 1=19$.

Step4: Find the length of AB

We know that $AC=AB + BC$. Given $AC = 43$ and $BC = 19$, then $AB=AC - BC$. So $AB=43-19 = 24$.

Answer:

24