Question Video: Determining If a Circle Can Pass through Four Given Points Using the Properties of Cyclic Quadrilaterals Mathematics

Given that 𝑚∠𝐵𝐶𝐴 = 61°, and 𝑚∠𝐷𝐴𝐵 = 98°, can a circle pass through the points 𝐴, 𝐵, 𝐶, and 𝐷?

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Video Transcript

Given that the measure of angle 𝐵𝐶𝐴 equals 61 degrees and the measure of angle 𝐷𝐴𝐵 equals 98 degrees, can a circle pass through the points 𝐴, 𝐵, 𝐶, and 𝐷?

Remember, if there are a pair of congruent angles subtended by the same line segment and on the same side of it, then their vertices and the segment’s endpoints lie on a circle in which that segment is a chord. Well, we have a line segment 𝐵𝐴, from which angle 𝐵𝐶𝐴 and 𝐵𝐷𝐴 are subtended. The angles lie on the same side of that line segment. So if angle 𝐵𝐶𝐴 is equal to angle 𝐵𝐷𝐴, then all four of our points must lie on the circumference of a circle. Now, we’re given that the measure of angle 𝐵𝐶𝐴 is 61 degrees and the measure of angle 𝐵𝐴𝐷 is 98.

Since triangle 𝐵𝐷𝐴 is isosceles, we can calculate the measure of angle 𝐵𝐷𝐴 by subtracting 98 from 180 and then dividing by two. And that gives us that the measure of angle 𝐵𝐷𝐴 is 41 degrees. So we see that the measure of angle 𝐵𝐶𝐴 is not equal to the measure of angle 𝐵𝐷𝐴. Since these angles are not equal, we observe that a circle cannot pass through the points, and the answer is no.