2025 AMC 12A Problem 11

Problem 11 of 25IntermediateGeometry

The orthocenter of a triangle is the concurrent intersection of the three (possibly extended) altitudes. What is the sum of the coordinates of the orthocenter of the triangle whose vertices are

A(2, 31),

B(8, 27),

and

C(18, 27)?

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Solution

Since

B

and

C

both have

y = 27,

side

BC

is horizontal and the altitude from

A

is the vertical line

x = 2.

Side

AC

has slope

\dfrac{27 - 31}{18 - 2} = -\dfrac{1}{4},

so the altitude from

B

has slope

4

y - 27 = 4(x - 8).

x = 2,

y = 27 + 4(2 - 8) = 3.

The orthocenter is

(2, 3),

with coordinate sum

5.

Thus, the correct answer is A.

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