# Question Video: Find the Measure of an Angle in a Quadrilateral Using the Properties of Tangents and the Properties of Chords Mathematics

Given that line segment π΄π΅ is a tangent to the circle π at the point π΅, πβ π΅ππΆ = 3 πβ π΄, and the point πΆ is the midpoint of line segment π·πΈ, find the value of π₯.

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### Video Transcript

Given that line segment π΄π΅ is a tangent to the circle π at the point π΅, the measure of angle π΅ππΆ is equal to three times the measure of angle π΄, and the point πΆ is the midpoint of line segment π·πΈ, find the value of π₯.

First of all, letβs 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΅ππΆ is three times the measure of angle π΄. πΆ is the midpoint of π·πΈ. And by looking at our figure, we see one other angle measure. We see that the measure of angle π΅ππΆ is equal to three π₯. From these properties, letβs 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 ππ΅ 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 πΆ is the midpoint of line segment π·πΈ, weβre showing that the line from the center of the circle π to point πΆ bisects the chord πΈπ·. And when thatβs the case, it is a perpendicular bisector. There will be a right angle here. So we can say that the measure of angle ππΆπ΄ is 90 degrees. We can also say that π΄π΅ππΆ 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΅ππΆ is equal to three times the measure of angle π΄. And the measure of angle π΅ππΆ is three π₯ degrees.

If we plug in three π₯ degrees for the measure of angle π΅ππΆ, 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 weβll 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. Weβll 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, weβll divide by four on both sides of the equation to get π₯ by itself. And weβll see that π₯ degrees equals 45 degrees. Therefore, π₯ equals 45.