Two tangents XY and XZ are drawn from a point X to a circle with cente

Question

Two tangents XY and XZ are drawn from a point X to a circle with center W.If the length of XZ is 156 and the length YZ is 120,find the radius of the circle.

(A) 60
(B) 65
(C) 70
(D) 75
(E) 80

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GMATinsight · Accepted Answer

CONCEPT : 
- Two tangents drawn to a circle from the same exterior point are always equal in length
- Point of tangency joined with the center of circle makes 90º with tangent

i.e. XY = XZ = 156

∆XYZ is isosceles triangle with sides 156-156-120

Check next part of solution in image attached

Answer: Option B

Lipun · Answer

\frac{r}{156}=\frac{60}{\sqrt{(156^2 - 60^2)}}

Bunuel , any other time efficient approach?

Thanks
Lipun

yashikaaggarwal · Answer

Draw a line from x to W which is perpendicular to YZ at O. Such that we have four triangles. YWO , ZWO, YXO and ZXO.
We know ‹WYX and angle WZX is 90. (Angle drawn from centre to tangent.)
XY and XZ are equal. Tangent drawn from same outside point.
Also YZ is a chord.
The triangle WYX and WZX
YW = WZ ( radius)
XW common.
YX= ZX (tangents)
Therefore angle yxw = zxw -----(1)
Since XW cut YZ at 90°

In ∆ OYX and ∆ OZX 
yxw = zxw (proved above)
Angel xoy = xoz = 90
Xy = Xz ( tangents)
So ∆ OYX and ∆ OZX are equal, YO and ZO are equal = 60 each and right angle triangle
Using Pythagoras,
XY^2 = XO^2+YO^2
156^2-60^2=YO^2
YO = 144 ------(2)

Draw a triangle with YZ as base and tip on the circle. The third side be V
Angle OYX = angle YVZ (angel drawn between tangent and the chord is equal to the angle drawn by chord at the circle)
2* Angle YVZ = angle YWZ
(Angle at the centre is twice the angle at circle.)
Angle YWX = 2*(Angle YWO)

Now in triangle YWO and YXO
Angle OYX = angle YWO
Angle WXO is = angle OXY
YOW= YOX = 90
Both triangle are similar
Therefore using similar triangle property,
YO/OX = WY/XY
60/144 = WY/156
WY = 60*156/144 = 65
WY = 65 (radius)
Answer is B

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bidskamikaze · Answer

Answer 65
From similar triangle property. 
156/r = 144/60
r= 65

bumpbot · Answer

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Two tangents XY and XZ are drawn from a point X to a circle with cente

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