AMC 10 · 2012 · #11
Grade 8 geometry-2dExternally tangent circles with centers at points A and B have radii of lengths 5 and 3, respectively. A line externally tangent to both circles intersects ray AB at point C. What is BC?
Pick an answer.
AMC 10 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: 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.
Givens: Circle A has radius 5 and circle B has radius 3.; The circles are externally tangent, so they touch at one point and neither is inside the other.; A line is tangent to both circles (an external tangent that touches each circle once).; That tangent line meets ray AB at point C.
Unknowns: The distance BC from center B to the crossing point C.
Understand
Restated: 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.
Givens: Circle A has radius 5 and circle B has radius 3.; The circles are externally tangent, so they touch at one point and neither is inside the other.; A line is tangent to both circles (an external tangent that touches each circle once).; That tangent line meets ray AB at point C.
Plan
Primary tool: #1 Draw a Diagram
Secondary: #7 Identify Subproblems, #4 Introduce a Variable, #13 Convert to Algebra
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.
Execute — Answer: D
4.G.A.1 Step 1 Draw the picture
- Mark where the tangent line touches circle A (call it P) and circle B (call it Q).
- Draw radius AP and radius BQ.
- A radius meets a tangent at a right angle, so AP and BQ are both perpendicular to the same line, which makes AP parallel to BQ, with lengths 5 and 3.
- Since the circles are externally tangent, the centers are AB = 5 + 3 = 8 apart.
💡 Drawing the radii to the touch points is what exposes the right angles that make the rest of the problem work.
8.G.A.5 Step 2 Find the two triangles
- Look at triangle CAP (through the big circle) and triangle CBQ (through the small circle).
- Both contain the point C, so they share the angle at C.
- Each also has a right angle, at P and at Q.
- Two matching angles force the third to match too, so the triangles are the same shape (similar).
💡 When two triangles agree in two angles they must have the same shape, just at different sizes.
7.RP.A.2 Step 3 Read off the ratio
- Same-shape triangles have matching sides in the same ratio.
- The side BQ = 3 in the small triangle matches AP = 5 in the big triangle, and side CB matches side CA.
- So the ratio of these matching sides is the same: CB to CA is 3 to 5.
💡 In similar triangles every pair of matching sides shrinks by the exact same factor.
7.EE.B.4 Step 4 Name BC and place C
- Let BC = x.
- Because the small circle has the smaller radius, the tangent line leans toward it and crosses the line of centers past B, so the order along the ray is A, then B, then C.
- That means CA is the whole stretch from A to C, which is AB plus BC: CA = 8 + x.
💡 The tangent tilts toward the smaller circle, so the meeting point sits just beyond the smaller center.
7.EE.B.4 Step 5 Solve the equation
- Put the ratio and the lengths together: x over (8 + x) equals 3 over 5.
- Cross-multiplying gives 5x = 3(8 + x) = 24 + 3x.
- Subtract 3x from both sides to get 2x = 24, so x = 12.
- Therefore BC = 12, which is answer (D).
💡 Turning the side ratio into one equation lets a single line of algebra pin down BC.
4.G.A.1 Mark where the tangent line touches circle A (call it P) and circle B (call it Q 8.G.A.5 Look at triangle CAP (through the big circle) and triangle CBQ (through the smal 7.RP.A.2 Same-shape triangles have matching sides in the same ratio. The side BQ = 3 in t 7.EE.B.4 Let BC = x. Because the small circle has the smaller radius, the tangent line le 7.EE.B.4 Put the ratio and the lengths together: x over (8 + x) equals 3 over 5. Cross-mu Review
Reasonableness: 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.
Alternative: Use the external center of similitude directly: the two external tangents of two circles meet on line AB at the point that divides it externally in the ratio of the radii, 5 : 3. Setting CA : CB = 5 : 3 with CA - CB = AB = 8 gives CB(5/3 - 1) = 8, so CB(2/3) = 8 and CB = 12, matching the triangle method.
CCSS standards used (min grade 8)
4.G.A.1Draw points, lines, rays, angles, and perpendicular lines, and identify them in figures (Drawing the radii to the tangent points and recognizing that a radius is perpendicular to the tangent, creating right angles at P and Q.)8.G.A.5Use informal arguments including the angle-angle criterion for similarity of triangles (Arguing that triangles CBQ and CAP share the angle at C and each have a right angle, so by angle-angle they are similar.)7.RP.A.2Recognize and represent proportional relationships between quantities (Reading the equal ratio of matching sides from the similar triangles, CB/CA = BQ/AP = 3/5.)7.EE.B.4Use variables to represent quantities and construct simple equations to solve problems (Naming BC = x, writing CA = 8 + x, and solving the proportion x/(8+x) = 3/5 for x.)
⭐ 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 radii to where a line touches each circle, spot the matching right triangles, and a simple ratio of sides gives the length.
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