AMC 10 · 2011 · #18

Grade 8 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
isosceles-triangleangle-sum-trianglethirty-sixty-ninety-triangle identify-subproblems ↑ Prerequisites: angle-sum-triangle
📏 Medium solution 💡 2 insights
Problem
In rectangle ABCD the long side AB = 6 and the short side BC = 3. Point M lies on side AB, chosen so that ∠ AMD = ∠ CMD. Find the degree measure of ∠ AMD.

Pick an answer.

(A)
15
(B)
30
(C)
45
(D)
60
(E)
75

AMC 10 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

No picture is given, so the first move is to draw the rectangle and mark the two equal angles at M (Tool #1). The drawing exposes two facts you would otherwise miss: AB ∥ DC turns ∠ AMD into an alternate interior angle equal to ∠ MDC, and that makes triangle CMD isosceles. From there Tool #7 (Identify Subproblems) splits the work into a short isosceles-triangle step and a right-triangle step, and Tool #4 (Introduce a Variable) lets the straight-line angle sum finish it in one equation.

1STEP 1

Draw and label the rectangle

Put AB = 6 along the bottom and BC = 3 on the right, mark M on AB, draw MD and MC, and tick the two equal angles at M.

A=(0,0), B=(6,0), C=(6,3), D=(0,3), M=(m,0)
2STEP 2

Spot the alternate interior angles

DC is parallel to AB and MD crosses both, so ∠ AMD and ∠ MDC are alternate interior angles — equal.

AB ∥ DC → ∠ AMD = ∠ MDC
3STEP 3

Find the isosceles triangle

Chaining with the given ∠ AMD = ∠ CMD gives ∠ CMD = ∠ MDC, so triangle CMD is isosceles: CM = CD = 6.

∠ CMD = ∠ MDC → CM = CD = 6
4STEP 4

Read the right triangle at B

In right triangle CMB the leg CB = 3 is exactly half the hypotenuse CM = 6 — the 30-60-90 shape — so ∠ CMB = 30°.

MB=√(6²-3²)=3√3, 3:3√3:6 → ∠ CMB = 30°
5STEP 5

Add the angles along the line

With θ = ∠ AMD the three angles at M fill line AB: 2θ + 30° = 180°, so θ = 75° — choice (E).

2θ + 30° = 180° → θ = 75° = (E)
Answer
75
Check that ∠ AMD = 75° holds together. If ∠ CMB = 30° then MB = CB/tan 30° = 3√(3) ≈ 5.20, so M ≈ (0.80, 0) and CM = √((3√3)² + 3²) = √(27+9) = 6 = CD, confirming the isosceles triangle. The three angles at M are 75° + 75° + 30° = 180°, exactly a straight line. Everything closes, and 75° is choice (E).
💡Key takeaway

Parallel sides made the angle bounce back equal, turning a triangle isosceles; then a 30-60-90 corner and the straight-line total of 180° pinned the angle at 75°.

  • Draw and label the rectangle
  • Spot the alternate interior angles
  • Find the isosceles triangle
  • Read the right triangle at B
  • Add the angles along the line

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