Question

fluency

1. in the diagram below point c is the mid - point of $overline{ab}$. if the length of $overline{ab}$ is 32 and the length of $overline{ac}$ is given by $\frac{x}{2}+10$ then which of the following is the value of x?

(1) 11 (3) 3
(2) 12 (4) 44

1. points m, r, p, q, and n all lie on a straight line. point p is the mid - point of segment $overline{mn}$. as well, point q is the mid - point of segment $overline{pn}$. finally, point r is the mid - point of segment $overline{mq}$. if the length of $overline{mn}$ is 24 units, which of the following is the length of $overline{mr}$?

(1) 6 (3) 10
(2) 9 (4) 12

Explanation:

Step1: Recall mid - point property

Since C is the mid - point of $\overline{AB}$, then $AC=\frac{1}{2}AB$. Given $AB = 32$, so $AC=\frac{32}{2}=16$.

Step2: Set up equation

We know that $AC=\frac{x}{2}+10$. Substitute $AC = 16$ into the equation: $\frac{x}{2}+10 = 16$.

Step3: Solve for x

Subtract 10 from both sides: $\frac{x}{2}=16 - 10=6$. Multiply both sides by 2 to get $x = 12$.

Step1: Use mid - point property for $\overline{MN}$

Since P is the mid - point of $\overline{MN}$ and $MN = 24$, then $MP=PN=\frac{MN}{2}=\frac{24}{2}=12$.

Step2: Use mid - point property for $\overline{PN}$

Since Q is the mid - point of $\overline{PN}$, then $PQ=\frac{PN}{2}=\frac{12}{2}=6$.

Step3: Find MQ

$MQ=MP + PQ=12 + 6=18$.

Step4: Use mid - point property for $\overline{MQ}$

Since R is the mid - point of $\overline{MQ}$, then $MR=\frac{MQ}{2}=\frac{18}{2}=9$.

Answer:

(2) 12