Skip to main content

2025 AMC 12A Problem 11

Problem 11 of 25IntermediateAlgebraGeometry

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),A(2, 31), B(8,27),B(8, 27), and C(18,27)?C(18, 27)?

Answer choices

Show solution

Solution

Since BB and CC both have y=27,y = 27, side BCBC is horizontal and the altitude from AA is the vertical line x=2.x = 2. Side ACAC has slope 27−3118−2=−14,\dfrac{27 - 31}{18 - 2} = -\dfrac{1}{4}, so the altitude from BB has slope 44: y−27=4(x−8).y - 27 = 4(x - 8). At x=2,x = 2, y=27+4(2−8)=3.y = 27 + 4(2 - 8) = 3. The orthocenter is (2,3),(2, 3), with coordinate sum 5.5. Thus, the correct answer is A.
AoPS wiki

Tagged: coordinate geometry · altitude · slope

More practice