AMC 10 · 2012 · #10

Grade 10 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Pythagorean / Distance Formula for a Length
area-trianglesmedian-of-triangletrigonometric-ratiossine-area-formula identify-subproblemsconvert-to-algebra ↑ Prerequisites: area-trianglestrigonometric-ratios
📏 Medium solution 💡 2 insights
Problem
A triangle's area, one side, and the median to that side are all known. Find the sine of the angle they make.

Pick an answer.

(A)
$\frac{3}{10}$
(B)
$\frac{1}{3}$
(C)
$\frac{9}{20}$
(D)
$\frac{2}{3}$
(E)
$\frac{9}{10}$
How to solve
Strategy Draw a Diagram

The three numbers control three different things, and the whole problem is seeing which two of them meet at the angle. With base AB = 10, the area controls only one quantity: the distance from C to the line AB. The median controls a second: the distance from C to D. Sine of the angle at D is precisely the first divided by the second, so one auxiliary line — the perpendicular from C down to line AB — turns the problem into a right triangle whose legs are already known. That is tool #1. Finding the height first is the clean subproblem (tool #7). Two things the picture alone cannot settle still need proof: that a triangle with these givens exists at all, and that no choice of picture changes the answer. Coordinates (tool #4) build an explicit triangle, and re-reading the coordinate solution as a set of reflections (tool #15) shows every alternative picture is congruent to that one.

1STEP 1

Name the side, midpoint, and median

The midpoint halves the side.

AD = DB = 10/2 = 5, CD = 9, [ABC] = 30
2STEP 2

The area fixes the height

The area fixes the height at 6.

30 = 1/2 · 10 · h → h = 6
3STEP 3

The median is a hypotenuse

The median is the hypotenuse of that right triangle.

sinθ = CE/CD = 6/9 = 2/3
4STEP 4

Confirm such a triangle exists

Such a triangle really does exist.

|y| = 6, x² + y² = 81 → x² = 45 → x = ± 3√(5)
5STEP 5

No picture changes the answer

Every placement gives 2/3, choice (D).

C ∈ {(± 3√(5), ± 6)} → sinθ = 6/9 = 2/3
Answer
2/3
Every check agrees. A sine must lie between 0 and 1, and 2/3 does; it is strictly less than 1 exactly because the height 6 is strictly less than the median 9 — if the median were also 6 the angle would be a right angle and no acute angle would exist, and a median shorter than 6 would be impossible. The explicit vertex C = (3√(5), 6) reproduces all three givens, so the answer is not vacuous. The wrong choices are all ratios of the wrong pair of numbers: 9/10 is the median over the side, 9/20 is the median over twice the side, 3/10 is the height over twice the side, and 1/3 is half the height over the median. Only the height over the median — the two lengths that actually meet as leg and hypotenuse at θ — gives 2/3.
💡Key takeaway

The area tells you how far the far corner sits from the side; divide that distance by the median's length and you get the sine of the angle between them.

  • Name the side, midpoint, and median
  • The area fixes the height
  • The median is a hypotenuse
  • Confirm such a triangle exists
  • No picture changes the answer

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