Question Video: Finding the Measure of an Angle given the Measure of an Arc by Using the Properties of Tangents to the Circle | Nagwa Question Video: Finding the Measure of an Angle given the Measure of an Arc by Using the Properties of Tangents to the Circle | Nagwa

Question Video: Finding the Measure of an Angle given the Measure of an Arc by Using the Properties of Tangents to the Circle Mathematics • First Year of Secondary School

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Find ๐‘ฅ.

02:19

Video Transcript

Find ๐‘ฅ.

Letโ€™s look at the diagram weโ€™ve been given. There is a circle and then two lines ๐ด๐ต and ๐ด๐ถ which are ach tangents to the circle because they each intercept the circle in only one place. These two tangents intersect at a point outside the circle, point ๐ด. And weโ€™re told that the measure of the angle formed by their intersection is ๐‘ฅ. The other piece of information weโ€™re given in the diagram is the measure of the minor arc ๐ต๐ถ. Thatโ€™s the minor arc intercepted by these two tangents.

In order to calculate the value of ๐‘ฅ, we need to recall the angles of intersecting tangents theorem. This states that the measure of the angle formed by the intersection of two tangents outside a circle is half the positive difference of the measures of the intercepted arcs. We already said that the minor arc intercepted by these two tangents is the arc ๐ต๐ถ, whose measure weโ€™ve been given. The major arc intercepted by these two tangents is the major arc ๐ต๐ถ. And in order to distinguish between these two arcs, we can place a point ๐ท anywhere on the major arc. To obtain the positive difference, we need to subtract the measure of the minor arc from the measure of the major arc. So we have that ๐‘ฅ is equal to a half the measure of the arc ๐ต๐ท๐ถ minus the measure of the arc ๐ต๐ถ.

Weโ€™re given the measure of the arc ๐ต๐ถ, but what about the arc ๐ต๐ท๐ถ? We can work this out if we recall that the measure of a full circle is 360 degrees and the measures of the major and minor arcs must therefore sum to this value. Subtracting 151 degrees from 360 degrees then, we find that the measure of the major arc ๐ต๐ท๐ถ is 209 degrees. We have then that ๐‘ฅ is equal to a half of 209 degrees minus 151 degrees. Thatโ€™s a half multiplied by 58 degrees, which is 29 degrees. So by recalling the angles of intersecting tangents theorem, we found that the value of ๐‘ฅ, which is the measure of the angle formed by the intersection of two tangents outside a circle, is 29 degrees.

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