Question

if m is the midpoint of $overline{ab}$, $am = b + 6$, and $ab = 3b + 3$, find $mb$. (draw a picture.)

Explanation:

Step1: Use mid - point property

Since $M$ is the mid - point of $\overline{AB}$, we know that $AB = 2AM$.

Step2: Set up the equation

We are given $AM=b + 6$ and $AB=3b + 3$. Substituting into $AB = 2AM$, we get $3b+3=2(b + 6)$.

Step3: Solve the equation for $b$

Expand the right - hand side: $3b+3=2b + 12$. Subtract $2b$ from both sides: $3b-2b+3=2b-2b + 12$, which simplifies to $b+3=12$. Then subtract 3 from both sides: $b=12 - 3=9$.

Step4: Find $AM$

Substitute $b = 9$ into the expression for $AM$: $AM=b + 6=9+6 = 15$.

Step5: Find $MB$

Since $M$ is the mid - point of $\overline{AB}$, $MB=AM$. So $MB = 15$.

Answer:

15