Video Transcript
Line segments 𝐴𝐵 and 𝐴𝐶 are two
chords in the circle with center 𝑀 in two opposite sides of its center, where the
measure of angle 𝐵𝐴𝐶 is 33 degrees. If 𝐷 and 𝐸 are the midpoints of
the line segments 𝐴𝐵 and 𝐴𝐶, respectively, find the measure of angle 𝐷𝑀𝐸.
We begin by noticing that 𝑀𝐸 and
𝑀𝐷 both pass through the center of the circle and that they bisect the chords 𝐴𝐶
and 𝐴𝐵, respectively. We can therefore apply the chord
bisector theorem, which states if we have a circle with center 𝑀 containing a chord
𝐴𝐵, then the straight line which passes through 𝑀 and bisects 𝐴𝐵 is
perpendicular to 𝐴𝐵. This means that, on our diagram,
the measure of angle 𝑀𝐸𝐴 and the measure of angle 𝑀𝐷𝐴 are both equal to 90
degrees.
We notice that 𝐴𝐷𝑀𝐸 is a
quadrilateral. And we know that the angles in a
quadrilateral sum to 360 degrees. This means that the measure of
angle 𝐷𝑀𝐸 which we are trying to calculate is equal to 360 minus 90 minus 90
minus 33. This is equal to 147 degrees.
