AMC 10 · 2005 · #14

Grade 10 geometry-2d

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

tangent-line-to-circleisosceles-right-trianglecoordinate-geometry spatial-visualizationconvert-to-algebrasystems-of-equations ↑ Prerequisites: coordinate-geometryisosceles-right-trianglepythagorean-theorem

📏 Medium solution 💡 2 insights

Problem

A circle centred on the vertical axis touches two slanted lines through the origin and one horizontal line, crossing none of them. Find the radius.

Pick an answer.

(A)

$6\sqrt{2}-6$

(B)

(C)

$6\sqrt{2}$

(D)

(E)

$6+6\sqrt{2}$

How to solve

Strategy Draw a Diagram

A picture fixes what the words leave loose: the circle sits above the line y=6, wedged into the 90-degree corner made by y=x and y=-x. Once the picture is right, one fact does all the work: tangency means the distance from the center to the line is exactly the radius. Name the height k, write that distance three ways, and the geometry turns into one linear equation.

1STEP 1

Set the picture

Tangency means every distance from the centre equals the radius.

C=(0,k), k ＞ 6, radius=r

2STEP 2

The flat line gives one equation

The horizontal line gives the radius as a simple difference.

r = k - 6

3STEP 3

The slanted line gives a second

A forty-five degree triangle gives the slanted distance, and both slants agree.

r = k/√(2) = k√(2)/2

4STEP 4

Two expressions, one k

Setting the two expressions equal locates the centre.

k-6=k√(2)/2 ⟹ k·(2-√(2))/2=6 ⟹ k=12/(2-√(2))=(12(2+√(2)))/2=12+6√(2)

5STEP 5

Back to the radius

Subtracting gives 6+6√(2), choice (E).

r=k-6=(12+6√(2))-6=6+6√(2)≈ 14.49

Answer

6+6√(2)

Test the answer directly instead of trusting the algebra. With k=12+6√(2)≈ 20.485 and r=6+6√(2)≈ 14.485: the distance down to y=6 is 20.485-6=14.485=r, and the distance to y=x is 20.485/√(2)≈ 14.485=r. Both match, so all three lines really are tangent. A size check also rules out the small choices: the circle must reach from height 6 up past the center, so r=k-6 with k noticeably larger than 6; answers like 6√(2)-6≈ 2.49 or 6 would put the center at height 8.49 or 12, and at those heights the distance to y=x (6.0 and 8.49) does not equal the radius.

💡Key takeaway

Tangent is a distance statement: write the center-to-line distance for every tangent line, set them all equal to the radius, and the picture becomes equations.

Set the picture
The flat line gives one equation
The slanted line gives a second
Two expressions, one k
Back to the radius

Set the picture

The flat line gives one equation

The slanted line gives a second

Two expressions, one k

Back to the radius

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