Tangent circles

Geometry Level 4

Two circles are tangent to each other, and their diameters $d_1$ and $d_2$ satisfy $d_1 \times d_2=36$ .
Segment $AB$ is tangent to both circles. Find the length of $AB$ .

If you think that the length of $AB$ depends on the relative values of $d_1$ and $d_2$ , submit your answer as 9999.

The answer is 6.

2 solutions

Maria Kozlowska
Nov 14, 2016

Applying Pythagorean Theorem for triangle $CDE$ we get the following equation:

$(r_1+r_2)^2=(r_2-r_1)^2 + CE^2 \Rightarrow CE^2=4 \times r_1 \times r_2$

$AB=CE \Rightarrow AB=\sqrt{d_1 \times d_2}=\boxed{6}$ .

In other words, the length of the tangent segment for two externally tangent circles is a geometric mean of their diameters.

Note: This problem was inspired by A text-book of geometry by Wentworth , #235 on page 175.

Did the same, good problem +1

Jason Chrysoprase - 4 years, 7 months ago

The correct answer is 12. As noted in one other comment, (r1+r2)^2=(r2-r1)^2+(CE)^2 The derivative of that formula is equal to (CE)^2=4(r1)(r2) As we we given at the beginning, the product equals 36, so we can simplify the right side to 4(36), which equals 144. Taking the square root leaves you with 12.

Matthew Perry - 4 years, 6 months ago

Note the difference between the radius and the diameter.

Calvin Lin Staff - 4 years, 6 months ago

d=2r,so 6 is correct answer.

xiaopeng zhau - 4 years, 6 months ago

Best explaination

Rohit Pant - 2 years, 10 months ago

I would prefer to give that $r_1 \times r_2 = 9$ , so that $\sqrt{ 36 }$ isn't a quick guess. Do you mind if I make the edit?

Calvin Lin Staff - 4 years, 7 months ago

I do not mind. You can go ahead.

Maria Kozlowska - 4 years, 7 months ago

I meant this to be more of an educational problem than a challenge problem. That is a reason I asked the question in this way. Again, if you want it changed it is OK with me.

Maria Kozlowska - 4 years, 7 months ago

Ah ic. I will leave it in your original form then.

I think it's a great problem too :)

Calvin Lin Staff - 4 years, 7 months ago

Satyam Tripathi
Dec 12, 2016

Take 2 circles each of diameter 6

Short cut .....good approach

Rohit Pant - 2 years, 10 months ago

Tangent circles

The answer is 6.

2 solutions

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