$M$ is the midpoint of $\overline{AB}$ and $N$ is the midpoint of $\overline{AC}$, and $T$ is the intersection of $\overline{BN}$ and $\overline{CM}$, as shown. If $\overline{BN}\perp\overline{AC}$, $BN = 12$, and $AC = 14$, then find $CT$.

Question

ILoveMathAndILoveHelping · Accepted Answer

Using 2:1 ratios for medians of a triangle:
Line Segment NT is 4
Since we know line segment AC is 14 and n is the midpoint, line segment NC is 7.
Using Pythagorean theorem
7^2+4^2=x^2
49+16=x^2
x=Square root of 65

AoPS · Answer

DONT ASK QUESTIONS ASSIGNED IN CLASS

Istumbled · Answer

Hey, helpful hint- use the 2:1 ratio! And true, don’t post your hw problems

jeffbodman · Answer

dont post these pls

DenisGylaev · Answer

THANK YOU FOR TELLING ME THIS. I GOT THE NOTIFICATION THAT SOMEONE ANSWERED MY QUESTION 4 YEARS LATER. U JUST GAVE ME NOSTALGIA. <3 ILoveMathAndILoveHelping