Look at the circle you created that has point C (the midpoint of AB) as its center and passes through point A. What can you say about AB, angle BDA, angle BEA, and the measures of those two angles. Use what you know about the inscribed angle of a circle, and include your reasoning in your response.

The inscribed angle on the circumference of the circle subtended by the diameter at the center is a right angle

What can be said about segment AB is; Segment AB is the diameter of circle with midpoint at C

What can be said about angle BDA is; Angle BDA is an inscribed angle of the circle with midpoint at C, subtended by the diameter AB at the center

The measure of angle BDA is 90°

What can be said about angle BEA is; Angle BEA is an inscribed angle of the circle with midpoint at C, subtended by the diameter AB at the center

The measure of angle BEA is 90°

The given diagram shows;

Circle with midpoint C, along AB, and radius AC

The diameter of circle C = AB

Circle with midpoint A, intersecting with circle C at points D and E

Tangents from point B on circle, C, intersect with circle A at points D and E

Required parameters;

The segment AB is a line that intersects the circle C at two points and also passes through the the point C which is the center of the circle C

Therefore, the segment AB is the diameter of the circle with center at C

The angle BDA is the inscribed angle of circle C subtended by the points A and B which specifies the diameter of the circle with midpoint C

Therefore, the angle BDA is subtended by the diameter of the circle at the center

According to circle theorem, we have;

Angle subtended at the center = 2 × The angle subtended at the circumference

The angle subtended at the center by the diameter AB = 180° (Angle on a straight line)

Therefore;

The angle subtended at the center = 180° = 2 × Angle BDA

Angle BDA = 180°/2 = 90°

Angle BDA  = 90°

Similarly, we have;

The angle subtended at the center = 180° = 2 × Angle BEA

Angle BEA = 180°/2 = 90°

Angle BEA = 90°

Find out more about inscribed angles of a circle here:

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