AMC 10 · 2025 · #10
Grade 8 geometry-2dPick an answer.
The problem is pure position and shape, so the safest move is to draw it and pin it to a grid. Put the right angle at the origin so the two legs lie on the axes, and the altitude, the median, and their crossing point all become lines you can write with equations. Then the whole question turns into finding where two lines meet and reading off how that point splits the altitude. No clever trick is needed once the picture is on coordinates.
Name the parts of the triangle
Fix the angles and the shortest side.
Labeling each vertex by its angle keeps the shortest side, the altitude, and the median from getting mixed up.
7.G.A.2Draw A DiagramFind the 30-60-90 side lengths
The sides run one to root three to two.
Half of an equilateral triangle is exactly a 30-60-90, so the Pythagorean theorem hands you the root 3.
8.G.B.7Introduce A VariablePlace the triangle on a grid
Putting the right angle at the origin makes it easy.
Standing the right angle on the origin lets the two legs ride the axes, so coordinates fall out for free.
6.NS.C.6Draw A DiagramLocate the foot of the altitude
Intersecting gives the foot of the altitude.
Perpendicular slopes are negative reciprocals, so the altitude's equation writes itself once you have the hypotenuse's.
Perpendicular slopes are negative reciprocals, so the altitude's equation writes itself.
▸ Why?
A quarter turn sends a direction to one at right angles, which shows up as that flip and sign change.
▸ Why?
The triangle's fixed angles put its sides in known ratios, so the coordinates are exact from the start.
Write the median as a line
Write the median as a line.
A midpoint is just the average of the two endpoints, so the median's endpoints are easy to read off.
6.G.A.3Identify SubproblemsCross the two lines and split the altitude
Crossing the lines gives three sevenths.
Where two lines meet is one point that satisfies both equations, and that point's place along the altitude is the whole answer.
8.EE.C.8Convert To AlgebraWhen a shape question is all about where lines cross, drop it onto a grid and let the equations find the meeting point for you.
- Name the parts of the triangle
- Find the 30-60-90 side lengths
- Place the triangle on a grid
- Locate the foot of the altitude
- Write the median as a line
- Cross the two lines and split the altitude