2025 AMC 12A Problem 11

Question

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)?

5

17

10 + 4\sqrt{17} + 2\sqrt{13}

\dfrac{113}{3}

54

Answer: A

Concepts:coordinate geometry altitude slope

Difficulty rating: 1570

Small hint:

B

and

C

share

y = 27,

so the altitude from

A

is the vertical line

x = 2

Big hint:

The altitude from

B

is perpendicular to

AC

; intersect it with

x = 2

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).

At

x = 2,

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

The orthocenter is

(2, 3),

with coordinate sum

5.

Thus, the correct answer is A.