Home‎ > ‎Geometry‎ > ‎Geometry Questions of the Month‎ > ‎

February Question of the Month for Geometry

Point P is the circumcenter of triangle ABC.  If the measure of angle ABP is 40 degrees and the measure of angle CBP is 30 degrees, then what is the measure of angle PAC ?


Solution:

If point P is the circumcenter, then it is equidistant to all of the vertices of triangle ABC, so AP=BP=CP.  Therefore, all of the interior triangles are isosceles, so their base angles are congruent.  So if the measure of angle ABP is 40 degrees, then so is angle BAP.  Also, since the measure of angle CBP is 30 degrees, then so is angle PCB.  Since all of the angles of a triangle add up to 180 degrees, then angle APB measures 100 degrees and angle BPC measures 120 degrees.  Since there are 360 degrees in a circle, the measure of angle APC is 140 degrees.  Angles PAC and PCA have to be congruent and add up to 180 degrees along with angle APC, so the measure of angle PAC is 20 degrees.
Comments