Find the distance between the centroid of a triangle and a random line $l$

Question

We are given $\triangle ABC$ with centroid $M$ and a line $l$ such that $A \in l$ and $l$ does not intersect the segment $BC$. Find $d(M; l)=MM_1$ if $CC_1=24$ and $BB_1=10$. $B_1BCC_1$ is a right trapezoid. If $PP_1 \perp l, P \in l$, then $PP_1$ is the midsegment. This is what I have noticed so far. Would appreciate your help!

Answer 1

We can always assume line $(L)$ is on the $x$-axis and $C$ is on the $y$-axis $C_1$ is at the origin. Then, $C = (0, 24)$, $B = (h, 10)$ and $A = (k, 0)$ for some $h$ and $k$.

By formula of centroid, $$MM_1 = \dfrac {(y_A + y_B + y_C)}{3}= (24 + 10 + 0)/3 = 34/3.$$