AMC 10 · 2012 · #11

Grade 8 geometry-2d
Archetype Similar-Triangle Length / Area Ratio └ Sub-archetype Synthetic Similar-Triangle Length or Area
tangent-circlessimilar-trianglesratio-proportion physical-representationconvert-to-algebra ↑ Prerequisites: similar-triangles
📏 Medium solution 💡 2 insights
Problem
Two circles touch on the outside. One has center A and radius 5, the other has center B and radius 3. A straight line touches both circles on the outside and crosses ray AB at a point C. Find the length BC.

Pick an answer.

(A)
4
(B)
4.8
(C)
10.2
(D)
12
(E)
14.4

AMC 10 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

A tangent-line-and-circles problem is easiest once it is drawn: sketching the radii to the two touch points reveals two right triangles that share the corner at C. Those triangles are the same shape, so a ratio of matching sides turns the whole picture into one small equation for BC.

1STEP 1

Draw the picture

Mark the touch points P on circle A and Q on circle B; radii AP = 5 and BQ = 3 are both perpendicular to the tangent, and AB = 8.

AP = 5, BQ = 3, AP ⊥ PC, BQ ⊥ QC, AB = 8
2STEP 2

Find the two triangles

Triangles CAP and CBQ share the angle at C and each has a right angle, at P and at Q, so they are the same shape — similar.

∠ C = ∠ C, ∠ APC = ∠ BQC = 90° → △ CBQ ∼ △ CAP
3STEP 3

Read off the ratio

Matching sides of similar triangles share one ratio: CB matches CA just as BQ = 3 matches AP = 5, so CB : CA = 3 : 5.

CB/CA = BQ/AP = 3/5
4STEP 4

Name BC and place C

Let BC = x. The tangent leans toward the smaller circle, so C sits past B: the order is A, B, C and CA = 8 + x.

BC = x, CA = AB + BC = 8 + x
5STEP 5

Solve the equation

Cross-multiply x / (8 + x) = 3 / 5 to get 5x = 24 + 3x, so 2x = 24 and x = 12 — choice (D).

x/(8 + x) = 3/5 → 5x = 24 + 3x → 2x = 24 → x = 12
Answer
12
Check the ratio: with BC = 12 we get CA = 8 + 12 = 20, and 12/20 = 3/5, exactly the radius ratio, so the similar-triangle relationship holds. C also sits beyond B as expected for the smaller circle, and 12 is one of the listed choices, so answer (D) is consistent.
💡Key takeaway

Draw the radii to where a line touches each circle, spot the matching right triangles, and a simple ratio of sides gives the length.

  • Draw the picture
  • Find the two triangles
  • Read off the ratio
  • Name BC and place C
  • Solve the equation

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