Question

Segments $AD=10,$ $BE=6,$ $CF=24$ are drawn from the vertices of triangle $ABC,$ each perpendicular to a straight line $RS,$ not intersecting the triangle. Points $D,$ $E,$ $F$ are the intersection points of $RS$ with the perpendiculars. If $x$ is the length of the perpendicular segment $GH$ drawn to $RS$ from the intersection point $G$ of the medians of the triangle, then $x$ is:

$\dfrac{40}{3}$

$16$

$\dfrac{56}{3}$

$\dfrac{80}{3}$

undetermined

Answer: A

Concepts:centroid vector mean

Difficulty rating: 1480

Small Hint:

Signed distance from a point to a fixed line is an affine function

Big Hint:

The centroid is the average of the three vertices

Solution:

Because $RS$ does not intersect the triangle, the three perpendicular distances have the same sign. Signed distance to a fixed line is affine, and the centroid is the average of the vertices. Therefore its distance is the average $x=\frac{10+6+24}{3}=\frac{40}{3}.$

Therefore, the correct answer is A.

1967 AMC 12 Problem 37

Answer: A

Difficulty rating: 1480

Small Hint:

Big Hint:

Solution:

Problem 37 in Other Years