Two circles are externally tangent at T. The line AB is a common external tangent to the two circles, and P is the foot of the altitude from T to line AB. Find the length AB.

Rangcr897 Jun 23, 2024

#2**+1 **

We can set variables to complete this problem quite efiiciently.

First, let's set the center of the large circle as M and the center of the smaller circle as N.

Now, let's extend AB and and the line connecting the centers until they meet each other at a pont, O.

Let the distance of the left edge of the smaller circle and O be x.

Now, let's note that triangle BMO and ANO are similar. This is quite important.

From this, we can write

\(BM / MO = AN / NO \\ 4 / (4 + 2 + x) = 1 / ( 1 + x) \\ 4 / ( 6 + x) = 1 /(1 +x) \\ 4(1 + x) = 1 (6 + x) \\ 4 + 4x = 6 + x \\ 3x = 2 \\ x = 2/3\)

Now that we have the value for x, we can easily solve for AB. We get that

\(OA = \sqrt{ NO^2 - NA^2 } = \sqrt{ (1 + 2/3)^2 - 1^2} = \sqrt{ 25/9 -1} = \sqrt{16/9} = 4/3 \\ OB = \sqrt{ MO^2 - MB^2} = \sqrt{ (6 + 2/3)^2 - 4^2 } = \sqrt{ 400/9 -16} = \sqrt{256/9} = 16/3 \\ AB = OB = OA = 16/3 - 4/3 = 12/3 = 4 \)

So our answer is 4

Feel free to let me know if I messed up!

Thanks! :)

NotThatSmart Jun 23, 2024

#2**+1 **

Best Answer

We can set variables to complete this problem quite efiiciently.

First, let's set the center of the large circle as M and the center of the smaller circle as N.

Now, let's extend AB and and the line connecting the centers until they meet each other at a pont, O.

Let the distance of the left edge of the smaller circle and O be x.

Now, let's note that triangle BMO and ANO are similar. This is quite important.

From this, we can write

\(BM / MO = AN / NO \\ 4 / (4 + 2 + x) = 1 / ( 1 + x) \\ 4 / ( 6 + x) = 1 /(1 +x) \\ 4(1 + x) = 1 (6 + x) \\ 4 + 4x = 6 + x \\ 3x = 2 \\ x = 2/3\)

Now that we have the value for x, we can easily solve for AB. We get that

\(OA = \sqrt{ NO^2 - NA^2 } = \sqrt{ (1 + 2/3)^2 - 1^2} = \sqrt{ 25/9 -1} = \sqrt{16/9} = 4/3 \\ OB = \sqrt{ MO^2 - MB^2} = \sqrt{ (6 + 2/3)^2 - 4^2 } = \sqrt{ 400/9 -16} = \sqrt{256/9} = 16/3 \\ AB = OB = OA = 16/3 - 4/3 = 12/3 = 4 \)

So our answer is 4

Feel free to let me know if I messed up!

Thanks! :)

NotThatSmart Jun 23, 2024