1967 AMC 12 Problem 37

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.