The Tale of 2 Chords
Geometry
Level
1
As shown, a circle has 2 chords AB = CD.
Are the yellow and blue areas always equal?
No, not necessarily.
Yes, always!
No, never!
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1 solution
Let O be the center of the circle and E the intersection point of both chords.
If AB = CD, then the triangle OAB is congruent to OCD because of equal 3 sides; they are also isosceles triangles.
That makes the angle OAD = angle ODA = x and angle OCD = angle ODC = angle OBA = angle OAB = y, according to isosceles property.
Thus, arithmetically, the angle DCB = angle BAD = x+y, making the triangle AED also an isosceles triangle. That means AE = ED and EC = EB because AB = CD.
That makes the triangle AEC congruent to BED because angle AEC = angle BED, so AC = BD and the arc AC = arc BD.
As a result, the both colored portions are congruent because of all equal side lengths and will have the same area.
Note: The quadrilateral ABCD is an isosceles trapezoid because angle ABC = angle CDA due to angle of the same arc. Thus, AD // BC and AC = BD.