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touch AD at the same point E

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touch AD at the same point E

by sanju09 » Wed Mar 09, 2011 3:24 am
The triangle ABC has sides AB = 137, AC = 241 an BC = 200. There is a point D, on BC, such that both incircles of triangles ABD and ACD touch AD at the same point E. What is the length of CD?
A. 152
B. 174
C. 179
D. 183
E. 197



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by anshumishra » Wed Mar 09, 2011 10:57 am
sanju09 wrote:The triangle ABC has sides AB = 137, AC = 241 an BC = 200. There is a point D, on BC, such that both incircles of triangles ABD and ACD touch AD at the same point E. What is the length of CD?
A. 152
B. 174
C. 179
D. 183
E. 197



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Please refer to the diagram below :
This is based on the theory : Two tangents can always be drawn to a circle from any point outside the circle, and these tangents are equal in length

x+y = 137 --- 1
x+z = 241 ---- 2
y+z+2w = 200 --- 3

Need to find : CD = CF"+DF" = w+z = ?

Add (1) and (2) =>2x+y+z = 137+241 = 378 --- 4
Subtract (1) and (2) => z-y = 241-137 = 104 ---- 5

Add (3) and (5) => y+z+2w+z-y = 200+104
=> 2(z+w) = 304
=> z+w = 152, A

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by Night reader » Wed Mar 09, 2011 1:49 pm
:)
x^2-y^2=200^2 - 137^2;
x+y=241

(x-y)(x+y)=(200-137)(200+137)
x+y=241

x-y=200-137
x+y=241, 2x=304, x=152 --> CD=152
answer A
sanju09 wrote:The triangle ABC has sides AB = 137, AC = 241 an BC = 200. There is a point D, on BC, such that both incircles of triangles ABD and ACD touch AD at the same point E. What is the length of CD?
A. 152
B. 174
C. 179
D. 183
E. 197



[spoiler]https://gmatmaths.com/[/spoiler]
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