Given that 𝑥 degrees is the measure of the major arc 𝐵𝐶, find the value of 𝑥.
Let’s consider the diagram we’ve been given. We have a circle and two tangents to this circle, the lines 𝐴𝐵 and 𝐴𝐶. These two tangents intersect one another at a point outside the circle, point 𝐴. And we’re given the measure of the angle formed by their intersection. It’s 64 degrees. The value we need to find, that’s 𝑥, is the expression used for the measure of the major arc 𝐵𝐶. That is the major arc intercepted by these two tangents. And as it is a major arc, we know that its measure is greater than 180 degrees.
To answer this question, we need to recall the angles of intersecting tangents theorem. This tells us 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 know that the measure of the major arc 𝐵𝐶 is 𝑥 degrees. But what about the measure of the minor arc 𝐵𝐶?
Well, as the measure of a full circle is 360 degrees, an expression for the measure of the minor arc is 360 minus 𝑥 degrees. The positive difference in the measures of these two arcs will be found by subtracting the measure of the minor arc from the measure of the major arc. So we have an equation. 64 degrees is equal to one-half 𝑥 degrees minus 360 minus 𝑥 degrees.
As everything in this equation is measured in degrees, we can omit the units throughout. So we have 64 equals one-half of 𝑥 minus 360 minus 𝑥. And we can now solve this equation to find the value of 𝑥. We can multiply both sides of the equation by two, which will give 128 on the left-hand side, and at the same time distribute the negative sign over the inner set of parentheses. We have then 128 is equal to 𝑥 minus 360 plus 𝑥.
Next, we can group the like terms on the right-hand side to give 128 is equal to two 𝑥 minus 360. We can then add 360 to each side of this equation to give 488 is equal to two 𝑥 and finally divide both sides of the equation by two, giving 244 is equal to 𝑥.
Now, note that the measure of the major arc 𝐵𝐶 was given as 𝑥 degrees. So it’s correct that our value of 𝑥 is purely numeric and doesn’t include the degrees symbol. By recalling the angles of intersecting tangents theorem then, we found that the value of 𝑥 is 244.