AMC 10 · 2009 · #20

Grade 8 geometry-2d
Archetype Similar-Triangle Length / Area Ratio └ Sub-archetype Synthetic Similar-Triangle Length or Area
angle-bisector-theorempythagorean-theoremquadratic-equations convert-to-algebra ↑ Prerequisites: pythagorean-theorem
📏 Medium solution 💡 2 insights 📊 Diagram
Problem
In right triangle ABC the right angle is at B, with AB=1 and BC=2. The ray from A that cuts ∠ BAC into two equal angles hits side BC at a point D. Find the length BD.

Pick an answer.

(A)
$\frac {\sqrt3 - 1}{2}$
(B)
$\frac {\sqrt5 - 1}{2}$
(C)
$\frac {\sqrt5 + 1}{2}$
(D)
$\frac {\sqrt6 + \sqrt2}{2}$
(E)
$2\sqrt 3 - 1$

AMC 10 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

The Angle Bisector Theorem says the bisector from A divides the opposite side BC in the same ratio as the two sides that make the angle: BD/DC=AB/AC. To use it I first need the hypotenuse AC, so Tool #7 (Identify Subproblems) peels off a quick Pythagorean sub-task. The figure (Tool #1) shows the right angle and the bisector, confirming which sides form the ratio. Then Tool #4 (Introduce a Variable) names BD=x so the ratio becomes one clean equation I can solve.

1STEP 1

Find the hypotenuse AC

The right angle at B makes AC the hypotenuse, so AC²=1²+2²=5 and AC=√5.

AC=√(AB²+BC²)=√(1²+2²)=√5
2STEP 2

Set up the bisector ratio

The Angle Bisector Theorem gives BD/DC=AB/AC=1/√5. Let BD=x, so DC=2-x and the ratio becomes x/(2-x)=1/√5.

BD/DC=AB/AC=1/√5, x/(2-x)=1/√5
3STEP 3

Solve for x

Cross-multiplying gives √5 x=2-x, so x(√5+1)=2 and x=2/(√5+1).

√5 x=2-x → x(√5+1)=2 → x=2/(√5+1)
4STEP 4

Rationalize to match a choice

Multiply top and bottom by the conjugate √5-1: the denominator becomes 4, so BD=(√5-1)/2, choice (B).

x=2/(√5+1)·(√5-1)/(√5-1)=(2(√5-1))/4=(√5-1)/2 → (B)
Answer
(√5 - 1)/2
Numerically BD=(√5-1)/2≈(2.236-1)/2≈0.618, so DC≈1.382. Both are positive and add to 2, and BD ＜ DC, which fits the prediction that the bisector lands nearer the shorter side AB. Checking the ratio, BD/DC≈0.618/1.382≈0.447=1/√5, exactly the Angle Bisector Theorem ratio.
💡Key takeaway

An angle bisector splits the far side in the same ratio as the two sides that make the angle, so find that ratio, set the piece you want equal to x, and solve.

  • Find the hypotenuse AC
  • Set up the bisector ratio
  • Solve for x
  • Rationalize to match a choice

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