Just 2 lengths

Geometry Level 2

A B AB is the diameter of a circle.
P P is a point on A B AB extended such that P A = 4 PA = 4 .
T T is the tangent point from P P to the circle.
If P T = 6 PT = 6 , what is the diameter of the circle?

5 8 6 7

Calvin Lin by Calvin Lin

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2 solutions

Aditya Rao
Nov 3, 2016

According chord and tangent properties, [Let diameter A B = x AB=x ]
P T 2 = P A × P B ⇒ 36 = 4 ( 4 + x ) ⇒ x = 5 . {PT}^2=PA\times PB\Rightarrow 36=4(4+x)\Rightarrow x=\boxed{5}.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Nice usage of these properties :)

Calvin Lin Staff - 4 years, 7 months ago

Thanks a lot !

Aditya Rao - 4 years, 7 months ago
Roger Erisman
Nov 4, 2016

Let radius of circle = X and center of circle be O.

Triangle POT is a right triangle.

By Pythagorean's Theorem: X^2 + 6^2 = (X + 4)^2

X^2 + 36 = X^2 + 8X + 16

36 = 8X + 16

20 = 8X

2.5 = X = radius

5 = diameter

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Ah, nice way to bring in the pythagorean theorem by using thales' theorem

Calvin Lin Staff - 4 years, 7 months ago

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