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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.

 

 Jun 23, 2024

Best Answer 

 #2
avatar+1944 
+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/NO4/(4+2+x)=1/(1+x)4/(6+x)=1/(1+x)4(1+x)=1(6+x)4+4x=6+x3x=2x=2/3

 

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

OA=NO2NA2=(1+2/3)212=25/91=16/9=4/3OB=MO2MB2=(6+2/3)242=400/916=256/9=16/3AB=OB=OA=16/34/3=12/3=4

 

So our answer is 4

Feel free to let me know if I messed up!

 

Thanks! :)

 Jun 23, 2024
 #1
avatar+130081 
0

https://web2.0calc.com/questions/circles_146

 

 

cool cool cool

 Jun 23, 2024
 #2
avatar+1944 
+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/NO4/(4+2+x)=1/(1+x)4/(6+x)=1/(1+x)4(1+x)=1(6+x)4+4x=6+x3x=2x=2/3

 

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

OA=NO2NA2=(1+2/3)212=25/91=16/9=4/3OB=MO2MB2=(6+2/3)242=400/916=256/9=16/3AB=OB=OA=16/34/3=12/3=4

 

So our answer is 4

Feel free to let me know if I messed up!

 

Thanks! :)

NotThatSmart Jun 23, 2024
 #3
avatar+130081 
0

Good job, NTS !!!

 

cool cool cool

CPhill  Jun 23, 2024
 #4
avatar+1944 
+1

 

Thank you! :)

 

~NTS

NotThatSmart  Jun 23, 2024
edited by NotThatSmart  Jun 23, 2024
edited by NotThatSmart  Jun 23, 2024

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