AMC 10 · 2004 · #24

Grade 8 geometry-2d
Archetype Similar-Triangle Length / Area Ratio └ Sub-archetype Synthetic Similar-Triangle Length or Area
inscribed-anglesimilar-trianglesangle-bisector-theorem convert-to-algebraidentify-subproblems ↑ Prerequisites: similar-triangles
📏 Medium solution 💡 3 insights
Problem
Triangle ABC has side lengths AB=7, AC=8, BC=9. A circle is drawn through all three vertices. The bisector of angle BAC is extended until it hits that circle again at a point D. Find the value of AD/CD.

Pick an answer.

(A)
$\dfrac{9}{8}$
(B)
$\dfrac{5}{3}$
(C)
2
(D)
$\dfrac{17}{7}$
(E)
$\dfrac{5}{2}$

AMC 10 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

A picture is what unlocks this problem, so Tool #1 (Draw a Diagram) leads. Drawing the circle with the four points A, B, D, C on it shows the single fact everything hinges on: because AD splits angle A into two equal halves, the two arcs BD and CD are equal, so the chords BD and CD are equal. That turns two unknown lengths into one. Tool #4 (Introduce a Variable) names that shared length x, and Tool #7 (Identify Subproblems) supplies the one clean relation tying the four sides and the diagonal AD together (Ptolemy's relation for a cyclic quadrilateral). The variable x cancels at the end, which is exactly why the answer is a plain number.

1STEP 1

Draw the circle and read off equal chords

On the circle the order is A, B, D, C; the equal angles at A face equal arcs BD and CD, so BD = CD.

∠ BAD = ∠ CAD → arc BD = arc CD → BD = CD
2STEP 2

Name the shared length

Let CD = x, so BD = x too. Quadrilateral ABDC then has AB = 7, AC = 8, BD = x, CD = x, with diagonals AD and BC = 9.

CD = x, BD = x, AB = 7, AC = 8, BC = 9
3STEP 3

Tie the sides together with Ptolemy

Ptolemy on ABDC gives AD · BC = AB · CD + AC · BD, so AD · 9 = 7x + 8x = 15x and AD = 5x/3.

AD · 9 = 7x + 8x = 15x → AD = 15x/9 = 5x/3
4STEP 4

Form the ratio and watch x cancel

AD/CD = (5x/3)/x, and x divides out, leaving the ratio 5/3 — choice (B).

AD/CD = (5x/3)/x = 5/3 (B)
Answer
5/3
The answer 5/3 is a clean number with no leftover x, which is the right shape for a ratio of two lengths on the same figure. A quick size check backs it up: computing the real lengths gives cos A = (7²+8²-9²)/(2 · 7 · 8) = 2/7, from which CD ≈ 5.61 and AD ≈ 9.35, and 9.35 / 5.61 = 1.667 = 5/3. It also passes a symmetry sniff test: if the triangle were isosceles with AB = AC, the bisector would be a line of symmetry, D would sit opposite the midpoint of BC, and Ptolemy would give AD = 2 AB/BC CD — the same structure with 7+8 replaced by 2 AB.
💡Key takeaway

Cutting angle A in half aims the line at the middle of the far arc, so BD = CD; then one circle rule (Ptolemy) turns all the sides into AD = 5/3CD, and the ratio is 5/3.

  • Draw the circle and read off equal chords
  • Name the shared length
  • Tie the sides together with Ptolemy
  • Form the ratio and watch x cancel

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