A geometry problem by Michael Spirnak

Geometry Level 3

Draw a circle and construct a diameter of the circle. Label the endpoints of the diameter A and B. Plot any point on the boundary of the circle, but NOT on one of the endpoints of the diameter. Label that point P. Draw chords connecting point P to each of the endpoints of the diameter. Which of the following is true?

( A P ) 2 + ( B P ) 2 = ( A B ) 2 (AP)^{2}+(BP)^{2}=(AB)^{2} Angle APB measures less than 90º. Triangle APB is an obtuse triangle. A P = A B − B P 2 AP=\frac{AB-BP}{2}

Michael Spirnak by Michael Spirnak , 25, USA

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1 solution

Kay Xspre
Nov 23, 2015

Writing the illustration will results into the angle APB being right angle (exactly 90 degrees), as it is the angle inscribed in the semicircle, thus, Choice 1 and 4 is wrong. As the triangle APB being right triangle, we will get that ( A P ) 2 + ( B P ) 2 = ( A B ) 2 (AP)^2+(BP)^2=(AB)^2

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