A well-drawn diagram can suggest a method, but an exact geometry solution needs relationships that follow from the conditions. A useful bridge from AMC 12 practice is to make every auxiliary length and assumption explicit.
This original triangle exercise develops that habit. It is not a released AIME problem or a claim that every AIME geometry question can be reduced to one formula.
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Name the requested quantity before choosing a method
A triangle has side lengths 13, 14 and 15. Find its inradius. If the area is K and semiperimeter is s, decomposing the triangle into three triangles from the incenter gives K = rs.
Here s = 21. The task is therefore to find the area exactly, not to estimate a radius from the appearance of a sketch.
Route 1: use a suitable area identity
Heron's formula gives K2 = 21(21 - 13)(21 - 14)(21 - 15) = 21 × 8 × 7 × 6 = 7056. Thus K = 84.
The inradius is 84/21 = 4. The positive square root is appropriate because area is positive. Keep the exact value rather than rounding partway through the calculation.
Route 2: build a coordinate model
Place the side of length 14 on the x-axis from (0,0) to (14,0). Let the third vertex be (x,h), above the axis, with distances 13 and 15 to those endpoints.
Then x2 + h2 = 169 and (14 - x)2 + h2 = 225. Subtracting eliminates h and gives x = 5. Hence h = 12, so K = (1/2)(14)(12) = 84 again.
Use the second route as a structural check
The altitude foot is five units from one endpoint and nine from the other. Those distances and height 12 reconstruct the 13 and 15 side lengths. The model checks the geometry instead of merely agreeing on a final number.
In another triangle, an altitude foot might lie outside the base segment. Coordinates can handle that, but a sketch that silently assumes an internal foot can mislead.
Practise choosing rather than collecting formulas
After solving, ask which given information made each route efficient. Heron's formula uses the three sides directly; coordinates are helpful when distances produce equations that simplify.
In a new problem, choose from the structure rather than forcing the last method you learned. To discuss a geometry bridge, scan the QR code and send AMC12 AIME GEOMETRY with your own labelled diagram and the equation you could not justify. Use official resources for current participation requirements.
Continue reading: AIME Mock Practice After AMC 12: Plan for Separate Sessions.
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