Tag Archives: angle bisector

Equilateral Triangle and Square

A pentagon made up of equilateral triangle with side length on top of square is inside a circle passing through points , and . Find the radius of the circle. Source: NCTM Mathematics Teacher, August 2006 Solution Let be the … Continue reading

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Inscribed Circle of Equilateral Triangle

The circumference of a circle circumscribing an equilateral triangle is units. Find the number of square units in the area of the circle inscribed in the same triangle. Source: NCTM Mathematics Teacher, February 2006 SOLUTION In equilateral triangle , angle … Continue reading

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Small Circle

A circle is inscribed in a sector of a circle of radius (as shown). What is the radius of the small circle? Source: NCTM Mathematics Teacher, February 2006 SOLUTION is the angle bisector of because is equidistant from and . Triangle … Continue reading

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Three Tangent Circles

Three mutually tangent circles of radius one are surrounded by a larger circle that is simultaneously tangent to all three. What is the radius of the larger circle? Source: http://www.mathcircles.org SOLUTION The centers of the three mutually tangent circles form … Continue reading

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Regions in a Square

In the figure below, quadrilateral  is a square, and is the midpoint of the side . How do the areas of regions I, II, III, and IV compare? Another way to think about this is to consider the question, What are … Continue reading

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Inscribed in a Circle

An equilateral triangle and regular hexagon are inscribed in the same circle.   Find the ratio of the area of the triangle to that of the hexagon. Source: mathcontest.olemiss.edu 12/7/2009 SOLUTION First, rotate the equilateral triangle around center of the circle … Continue reading

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