Question

de bisects ac at point b. ac = 3x + 32 and ab = 3x + 1. find bc.

Explanation:

Step1: Recall bisect property

If a line bisects a line - segment, it divides the line - segment into two equal parts. So, $AB = BC$ and $AC=AB + BC$. Since $AB = BC$, we can also write $AC = 2AB$.

Step2: Set up the equation

Substitute the given expressions for $AC$ and $AB$ into the equation $AC = 2AB$. We have $3x + 32=2(3x + 1)$.

Step3: Expand the right - hand side

Using the distributive property $a(b + c)=ab+ac$, we get $3x + 32=6x + 2$.

Step4: Solve for $x$

Subtract $3x$ from both sides: $32=6x + 2-3x$, which simplifies to $32 = 3x+2$. Then subtract 2 from both sides: $32 - 2=3x$, so $30 = 3x$. Divide both sides by 3: $x = 10$.

Step5: Find $BC$

Since $BC = AB$ and $AB = 3x + 1$, substitute $x = 10$ into the expression for $AB$. So $AB=3\times10 + 1=30 + 1=31$. Since $BC = AB$, $BC = 31$.

Answer:

$31$