In the figure above, O is the center of the circle. Line AB intersects

Question

https://gmatclub.com/forum/download/file.php?id=38806 In the figure above, O is the center of the circle. Line AB intersects the circle only at point B, and line DC intersects the circle only at point C. If the circle has radius of 2, then AC = (A) 4 (B) 2√2 (C) 4 + √2 (D) 4 + √3 (E) 2 + 2√2 2017-11-23_2025_001.png

Vorovski · Accepted Answer

ABO is a right isosceles triangle, hence  BO = AB = r = 2 , hence the side AO =  2\sqrt{2} .  AO + r = 2\sqrt{2} + 2 . The answer is E

RicAssassin · Answer

E.

Produce OB and it shall make a right angle at point of contact.
Triangle ABO will be right and isosceles.

AO can be found from pythagoras theorem.

AO+radius is the answer.

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KarishmaB · Answer

OB is the radius of the circle and AB is the tangent. Radius is perpendicular to tangent at the point of intersection. In triangle OBA, angle A is 45 degrees, angle ABO is 90 degrees and hence angle BOA is also 45 degrees. This is a 45-45-90 triangle in which ratio of the sides will be  1:1:\sqrt{2} 
Using pythagorean theorem, AO  = 2\sqrt{2}

AC = AO + OC =  2\sqrt{2} + 2

Answer (E)

generis · Answer

ccccc.png The length of AC = (length of hypotenuse, AO, of an isosceles right triangle) + (radius, CO) Find length of AO Connect O with B Point B is tangent, so the radius is perpendicular to line AB and creates a right angle at point B The triangle hence is a 45-45-90 isosceles right triangle, with side lengths in ratio* x: x: x\sqrt{2} One leg, BO = x = radius = 2 Other leg, AB = BO = x = r = 2 Hypotenuse, AO = x\sqrt{2}= 2\sqrt{2} Add segment CO to get length of AC The rest of the length of AC is CO = r = 2 Length of AC = 2\sqrt{2} + 2 Answer E * OR Pythagorean theorem: leg^2 + leg^2 = hypotenuse, h^2 2^2 + 2^2 = h^2 8 = h^2 \sqrt{4*2} = \sqrt{h^2} h = 2\sqrt{2} = length of AO

In the figure above, O is the center of the circle. Line AB intersects

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In the figure above, O is the center of the circle. Line AB intersects

Prep Toolkit

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