### Video Transcript

Given that line segment π΄π΅ is a tangent to the circle π at the point π΅, the measure of angle π΅ππΆ is equal to three times the measure of angle π΄, and the point πΆ is the midpoint of line segment π·πΈ, find the value of π₯.

First of all, letβs list out the information we were given. We know that π΄π΅ is tangent to the circle at point π΅. We know that the measure of angle π΅ππΆ is three times the measure of angle π΄. πΆ is the midpoint of π·πΈ. And by looking at our figure, we see one other angle measure. We see that the measure of angle π΅ππΆ is equal to three π₯. From these properties, letβs see if we can draw any conclusions. Because π΄π΅ is tangent to this circle at point π΅ and point π is the center of the circle, the radius ππ΅ will be creating a right angle with the line segment π΄π΅. The radius is perpendicular to the tangent at point π΅. We can then say that the measure of angle π΄π΅π is 90 degrees.

After that, because πΆ is the midpoint of line segment π·πΈ, weβre showing that the line from the center of the circle π to point πΆ bisects the chord πΈπ·. And when thatβs the case, it is a perpendicular bisector. There will be a right angle here. So we can say that the measure of angle ππΆπ΄ is 90 degrees. We can also say that π΄π΅ππΆ is a quadrilateral, which means the interior angles must sum to 360 degrees. At this point, it seems like we know something about three of the four angles inside this quadrilateral. But then we remember that the measure of angle π΅ππΆ is equal to three times the measure of angle π΄. And the measure of angle π΅ππΆ is three π₯ degrees.

If we plug in three π₯ degrees for the measure of angle π΅ππΆ, we get the equation three π₯ degrees is equal to three times the measure of angle π΄. By dividing both sides of this equation by three, we can see that the measure of angle π΄ will be equal to π₯ degrees. And so weβll add that to our graph so that we can create the equation 90 degrees plus three π₯ degrees plus 90 degrees plus π₯ degrees will equal 360 degrees. Weβll combine like terms so that we have 180 degrees plus four π₯ degrees equals 360 degrees. If we subtract 180 degrees from both sides, we find out that four π₯ degrees equals 180 degrees. From there, weβll divide by four on both sides of the equation to get π₯ by itself. And weβll see that π₯ degrees equals 45 degrees. Therefore, π₯ equals 45.