Video: Finding the Angle of a Point outside a Circle given the Angle of the opposite Arc

Find ๐‘ฅ.

02:00

Video Transcript

Find ๐‘ฅ.

Letโ€™s have a look at the diagram weโ€™ve been given more closely. It consists of a circle and then two lines ๐ด๐ถ and ๐ด๐ธ. Each of these lines intersect with the circle in two places, which means the lines ๐ด๐ถ and ๐ด๐ธ are called secants. The value of ๐‘ฅ, which weโ€™ve been asked to find, is the measure of the angle formed where these two secants meet.

Weโ€™ve also been given the measures of the two intercepted arcs formed by these two secants. The measure of the minor arc ๐ต๐ท is 36 degrees. And the measure of the major arc ๐ถ๐ธ is 132 degrees. We need to recall the relationship that exists between the angle formed by two secants and the measures of the two intercepted arcs. We recall then that โ€œif two secants intersect outside a circle, the measure of the angle formed is half the difference of the measures of the intercepted arcs.โ€

Our two secants do intersect at a point outside the circle. They intersect at the point ๐ด. So we could apply this result. The measure of the angle formed is the value ๐‘ฅ that weโ€™re looking to find. Weโ€™re told that this is equal to half the difference of the measures of the intercepted arcs. So this is a half of the measure of the arc ๐ถ๐ธ minus the measure of the arc ๐ต๐ท, both of which weโ€™ve been given.

๐‘ฅ is therefore equal to a half of 132 degrees minus 36 degrees. 132 minus 36 is 96. So ๐‘ฅ is equal to a half of 96 degrees or a half times 96 degrees. A half of 96, which we can find by dividing 96 by two, is 48.

So we found that the value of ๐‘ฅ, which is the measure of the angle formed between the two secants ๐ด๐ถ and ๐ด๐ธ, is 48 degrees.

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