Question Video: Finding the Measure of the Two Arcs Inscribed between Secants Given the Inscribed Angle | Nagwa Question Video: Finding the Measure of the Two Arcs Inscribed between Secants Given the Inscribed Angle | Nagwa

# Question Video: Finding the Measure of the Two Arcs Inscribed between Secants Given the Inscribed Angle Mathematics • Third Year of Preparatory School

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Given that, in the shown figure, π¦ = (π₯ β 2) and π§ = (2π₯ + 2), determine the value of π₯.

03:15

### Video Transcript

Given that, in the shown figure, π¦ equals π₯ minus two and π§ equals two π₯ plus two, determine the value of π₯.

Letβs have a look at the diagram. Weβve been given a circle. We see that the line segment π΄π΅ is a tangent to this circle because it intersects the circle in only one place. The line segment π΄π· is a segment of a secant of a circle because it intersects the circle in two places. π΄π΅ and π΄π· intersect at a point outside the circle. And weβve been given the measure of the angle between them.

Weβve also been given expressions for the two intercepted arcs. The arc π΅πΆ has a measure of π¦ degrees, and the arc π΅π· has a measure of π§ degrees. But given in the question, we also have expressions for each of these variables in terms of the unknown π₯, which is the variable whose value we need to find. We can therefore add these expressions in terms of π₯ to the diagram.

Now, to answer this problem, we need to recall the theorem concerning the angle between a tangent and a secant which intersect outside a circle. This states that the measure of the angle formed by a secant and a tangent that intersect at a point outside a circle is half the positive difference of the measures of the intercepted arcs. The angle measure is 50 degrees. The intercepted arcs are the arcs π΅π· and π΅πΆ. And we can see from the diagram that the arc π΅π· has the larger measure, so we can form an equation. 50 degrees is equal to one-half, the measure of the arc π΅π· minus the measure of the arc π΅πΆ.

But remember, weβve been given expressions for each of these arc measures in terms of π₯. The measure of the arc π΅π·, which was π§ degrees, is two π₯ plus two degrees. And the measure of the arc π΅πΆ, which was π¦ degrees, is π₯ minus two degrees. So we have an equation in π₯. 50 degrees is equal to a half two π₯ plus two degrees minus π₯ minus two degrees. And in fact, as everything in this equation is measured in degrees, we donβt need to include the degrees symbol throughout.

Now, we can solve this equation for π₯. The first step is to multiply both sides by two, which will give 100 on the left-hand side and eliminate the fraction on the right-hand side. At the same time, we can also distribute the negative over the second set of parentheses. And we have the equation 100 equals two π₯ plus two minus π₯ plus two. Next, we group like terms on the right-hand side. Two π₯ minus π₯ is π₯ and two plus two is four. So our equation becomes 100 equals π₯ plus four. We can solve this equation in one step by subtracting four from each side, and it gives 96 is equal to π₯.

So by recalling the angles between intersecting secants and tangents theorem and then forming and solving an algebraic equation, we found that the value of π₯ is 96.

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