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2010 AMC 10B Problem 6

Problem 6 of 25EasierGeometry

A circle is centered at O,O, AB‾\overline{AB} is a diameter and CC is a point on the circle with ∠COB=50∘.\angle COB = 50^\circ. What is the degree measure of ∠CAB?\angle CAB?

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Solution

Consider the following diagram: We have that ∠AOC=180∘−50∘=130∘. \angle AOC = 180^{\circ} - 50^{\circ} = 130^{\circ}. Since △AOC\triangle AOC is isosceles, we have that ∠CAO=180∘−130∘2=25∘. \angle CAO = \dfrac{180^{\circ} - 130^{\circ}}{2} = 25^{\circ}. Since ∠CAB=∠CAO,\angle CAB = \angle CAO, we have that ∠CAB=25∘.\angle CAB = 25^{\circ}. Thus, B is the correct answer.

More practice

Concepts: circle · isosceles triangle · angle chasing

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.