AMC 10 · 2014 · #22

Grade 8 geometry-2d
Archetype Similar-Triangle Length / Area Ratio └ Sub-archetype Synthetic Similar-Triangle Length or Area
pythagorean-theoremthirty-sixty-ninety-triangle identify-subproblemsconvert-to-algebra ↑ Prerequisites: pythagorean-theorem
📏 Medium solution 💡 2 insights
Problem
In rectangle ABCD the long sides AB and CD measure 20 and the short sides BC and AD measure 10. Point E sits on the top side CD so that ∠ CBE=15°. Find the length of AE.

Pick an answer.

(A)
$\dfrac{20\sqrt3}3$
(B)
$10\sqrt3$
(C)
18
(D)
$11\sqrt3$
(E)
20

AMC 10 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

The whole problem lives in one picture, so Tool #1 (Draw a Diagram) is the anchor: drawing the rectangle and dropping E onto the top side reveals two right triangles hiding inside. Tool #7 (Identify Subproblems) splits the work cleanly — first the small right triangle BCE that hides the 15° angle, then the right triangle ADE that actually contains AE. Tool #4 (Introduce a Variable) names the unknown gap CE so the 15° angle can pin it down. Tool #3 (Eliminate Possibilities) closes it out by matching the clean final length to one of the five listed choices.

1STEP 1

Set coordinates and spot two right triangles

Put the rectangle on axes with E=(x,10) on the top side; right triangles BCE and ADE appear at C and D, and AE is ADE's hypotenuse.

A=(0,0), B=(20,0), C=(20,10), D=(0,10), E=(x,10)
2STEP 2

Use the 15 degree angle to find CE and DE

In triangle BCE the 15° angle at B gives CE=10 tan 15°=20-10√3, so the rest of the top side is DE=10√3.

CE=10tan15°=10(2-√3)=20-10√3, DE=20-CE=10√3
3STEP 3

Pythagorean theorem on triangle ADE

In right triangle ADE the legs are AD=10 and DE=10√3, so squaring and adding gives AE²=100+300=400.

AE²=AD²+DE²=10²+(10√3)²=100+300=400
4STEP 4

Take the square root and match a choice

Since 400 is a perfect square, the root is clean: AE=√400=20, which is exactly choice (E).

AE=√(400)=20 (E)
Answer
20
The answer 20 has to be believable as a slanted length inside the rectangle. It is longer than the short side AD=10 and shorter than the full diagonal AC=√(20²+10²)=√(500)≈22.4, so 20 fits the window. It also fits the picture: CE=20-10√3≈2.7 is small, so E sits close to C, which means AE should be nearly as long as the corner-to-corner diagonal AC≈22.4 — and 20 is indeed just under it. Everything is consistent, and the answer is (E).
💡Key takeaway

The 15° angle sets CE=20-10√3, so DE=10√3; then the right corner at D makes AE=√(10²+(10√3)²)=√(400)=20.

  • Set coordinates and spot two right triangles
  • Use the 15 degree angle to find CE and DE
  • Pythagorean theorem on triangle ADE
  • Take the square root and match a choice

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