Video Transcript
Line segments π΄π΅ and π΄πΆ are two
chords in the circle with center π in two opposite sides of its center, where the
measure of angle π΅π΄πΆ is 33 degrees. If π· and πΈ are the midpoints of
the line segments π΄π΅ and π΄πΆ, respectively, find the measure of angle π·ππΈ.
We begin by noticing that ππΈ and
ππ· both pass through the center of the circle and that they bisect the chords π΄πΆ
and π΄π΅, respectively. We can therefore apply the chord
bisector theorem, which states if we have a circle with center π containing a chord
π΄π΅, then the straight line which passes through π and bisects π΄π΅ is
perpendicular to π΄π΅. This means that, on our diagram,
the measure of angle ππΈπ΄ and the measure of angle ππ·π΄ are both equal to 90
degrees.
We notice that π΄π·ππΈ is a
quadrilateral. And we know that the angles in a
quadrilateral sum to 360 degrees. This means that the measure of
angle π·ππΈ which we are trying to calculate is equal to 360 minus 90 minus 90
minus 33. This is equal to 147 degrees.