AMC 10 · 2021 · #17

Grade 8 geometry-2d

Archetype Compound Area by Decomposition / Difference └ Sub-archetype Polygon Triangulation

similar-trianglespythagorean-theoremisosceles-triangleperpendicular-bisector identify-subproblemsconvert-to-algebra ↑ Prerequisites: similar-triangles

📏 Long solution 💡 3 insights 📊 Diagram

Problem

In trapezoid ABCD, the bases satisfy AB ∥ CD, the slanted sides give BC = CD = 43, and the diagonal BD is perpendicular to side AD. The diagonals AC and BD meet at O, and P is the midpoint of BD. Given OP = 11, write AD = m√(n) in simplest radical form and find m + n.

Pick an answer.

(A)

(B)

132

(C)

157

(D)

194

(E)

215

AMC 10 2021 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

The wording packs four facts (parallel sides, isosceles, perpendicular, midpoint) into one figure — Tool #1 (Diagram) keeps them straight and reveals the key right triangle △ CPD. Tool #7 (Subproblems) splits the question into three smaller ones: (a) find AB via similar triangles △ BDA ∼ △ BPC; (b) find BD via the diagonal ratio BO:OD = AB:CD = 2:1 combined with OP = 11; (c) find AD via Pythagorean theorem on △ ADB. Tool #13 (Algebra) handles the linear equation x/2 = 11 that ties step (b) together.

1STEP 1

BC = CD makes △BCD isosceles, so the median from C to midpoint P of BD is also the altitude: CP ⊥ BD.

CP ⊥ BD, ∠ CPB = 90°

2STEP 2

AB ∥ CD gives ∠ABD = ∠BDC = ∠DBC, and both triangles have a right angle, so △BDA ∼ △BPC by AA.

△ BDA ∼ △ BPC

3STEP 3

Similarity ratio AB/BC = BD/BP = 2 (since P bisects BD), so AB = 86.

AB = 2 · 43 = 86

4STEP 4

Parallel bases split the diagonals BO:OD = AB:CD = 2:1; with OD = x, BD = 3x and order D, O, P, B give OP = x/2.

OP = x/2, where OD = x

5STEP 5

Set OP = 11: x/2 = 11 gives x = 22, so BD = 66.

x = 22, BD = 66

6STEP 6

Right angle at D: Pythagoras gives AD² = 86² - 66² = (20)(152) = 3040.

AD² = 86² - 66² = 20 · 152 = 3040

7STEP 7

3040 = 16·190 with 190 = 2·5·19 square-free, so AD = 4√(190): m = 4, n = 190, and m + n = 194 → (D).

AD = 4√(190), m + n = 194 → (D)

Answer

194

Quick sanity: AB = 86 and BD = 66, so AD = √(86² - 66²) ≈ √(3040) ≈ 55.1 — a positive length shorter than the hypotenuse 86, exactly as a leg should be. And 4√(190) ≈ 4 · 13.78 ≈ 55.1 matches. The square-free check on 190 = 2 · 5 · 19 confirms the form is minimal, so m + n = 194 is the intended choice (D).

💡Key takeaway

Once you spot the right angle inside the isosceles triangle △ BCD, the rest is plug-and-chug: a similar-triangle ratio gives AB = 86, the diagonal split BO:OD = 2:1 plus OP = 11 gives BD = 66, and Pythagoras on △ ADB gives AD = 4√(190), so m + n = (D) 194.

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