Given that, in the shown figure, 𝑦 equals 𝑥 minus two and 𝑧 equals two 𝑥 plus two, determine the value of 𝑥.
Let’s have a look at the diagram. We’ve been given a circle. We see that the line segment 𝐴𝐵 is a tangent to this circle because it intersects the circle in only one place. The line segment 𝐴𝐷 is a segment of a secant of a circle because it intersects the circle in two places. 𝐴𝐵 and 𝐴𝐷 intersect at a point outside the circle. And we’ve been given the measure of the angle between them.
We’ve also been given expressions for the two intercepted arcs. The arc 𝐵𝐶 has a measure of 𝑦 degrees, and the arc 𝐵𝐷 has a measure of 𝑧 degrees. But given in the question, we also have expressions for each of these variables in terms of the unknown 𝑥, which is the variable whose value we need to find. We can therefore add these expressions in terms of 𝑥 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 𝐵𝐷 and 𝐵𝐶. And we can see from the diagram that the arc 𝐵𝐷 has the larger measure, so we can form an equation. 50 degrees is equal to one-half, the measure of the arc 𝐵𝐷 minus the measure of the arc 𝐵𝐶.
But remember, we’ve been given expressions for each of these arc measures in terms of 𝑥. The measure of the arc 𝐵𝐷, which was 𝑧 degrees, is two 𝑥 plus two degrees. And the measure of the arc 𝐵𝐶, which was 𝑦 degrees, is 𝑥 minus two degrees. So we have an equation in 𝑥. 50 degrees is equal to a half two 𝑥 plus two degrees minus 𝑥 minus two degrees. And in fact, as everything in this equation is measured in degrees, we don’t need to include the degrees symbol throughout.
Now, we can solve this equation for 𝑥. 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 𝑥 plus two minus 𝑥 plus two. Next, we group like terms on the right-hand side. Two 𝑥 minus 𝑥 is 𝑥 and two plus two is four. So our equation becomes 100 equals 𝑥 plus four. We can solve this equation in one step by subtracting four from each side, and it gives 96 is equal to 𝑥.
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 𝑥 is 96.