8. if $overline{bd}congoverline{bc}$, $bd = 5x - 26$, $bc = 2x + 1$, and $ac = 43$, find $ab$.
8. if $overline{bd}congoverline{bc}$, $bd = 5x - 26$, $bc = 2x + 1$, and $ac = 43$, find $ab$.
Answer
Explanation:
Step1: Set up equation from congruent segments
Since $\overline{BD}\cong\overline{BC}$, we have $5x - 26=2x + 1$.
Step2: Solve 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 $AB$
Since $AC=AB + BC$ and $AC = 43$, $BC=19$, then $AB=AC - BC$. So $AB=43-19 = 24$.
Answer:
$x = 9$ $AB = 24$