Question Video: Finding the Measures of the Angle and the Major and Minor Arcs Inscribed between Two Tangents Mathematics • Third Year of Preparatory School

Video Transcript

In the given figure, find the values of 𝑥 and 𝑦.

Let’s have a look at the diagram we’ve been given. There is a circle with center 𝑀 and then two tangents to this circle: the lines 𝐴𝐵 and 𝐴𝐶. These two tangents intersect at a point outside the circle, point 𝐴. And we’re told that the measure of the angle formed between these two tangents is 𝑥 degrees.

We’re also given expressions for the measures of the two arcs intercepted by these tangents. The arc we can assume to be the minor arc, 𝐵𝐶, has a measure of two 𝑥 degrees. And the measure of the major arc 𝐵𝐶 is 𝑦 degrees. In order to find the values of these unknowns 𝑥 and 𝑦, we need to recall the relationship that exists between the angle between two tangents and the measures of their intercepted arcs.

We know that the measure of the angle between two tangents that intersect outside a circle is half the positive difference of the measures of the intercepted arcs. We can therefore form an equation. The measure of the angle between the two tangents is 𝑥 degrees. The major intercepted arc is 𝑦 degrees, and the minor intercepted arc is two 𝑥 degrees. So we have the equation 𝑥 degrees is equal to a half 𝑦 degrees minus two 𝑥 degrees.

Now as everything in this equation is measured in degrees, we can remove the units. And we have 𝑥 is equal to a half multiplied by 𝑦 minus two 𝑥. Now we can’t solve this equation because we have two unknowns and only one equation, but we can manipulate it a little.

We’ll begin by multiplying both sides by two to give two 𝑥 is equal to 𝑦 minus two 𝑥. And then we can add two 𝑥 to each side to give four 𝑥 is equal to 𝑦. Now as we’ve already said, we can’t solve this equation because we have two unknowns and only a single equation. But we have at least defined explicitly what the relationship is between 𝑦 and 𝑥. In order to find the values of 𝑥 and 𝑦 though, we need a second equation.

Well, we also know that the measure of a full circle is 360 degrees. Adding together the measures of the major and minor arcs then, we can form a second equation, 𝑦 plus two 𝑥 is equal to 360. We now have two linear equations in two unknowns. And so we can solve them simultaneously. Our first equation 𝑦 equals four 𝑥 gives an expression for 𝑦 in terms of 𝑥. So we can substitute this expression for 𝑦 into our second equation, giving four 𝑥 plus two 𝑥 equals 360.

We can then group the like terms on the left-hand side of the equation to give six 𝑥 equals 360 and finally divide both sides of the equation by six to give 𝑥 equals 60. We’ve found the value then of one of the two unknowns. In order to find the value of the other, we need to substitute this value of 𝑥 into the other equation. That gives 𝑦 equals four multiplied by 60, which is 240. A quick check confirms that this value of 𝑦 plus twice this value of 𝑥 is indeed equal to 360.

So by recalling the angles of intersecting tangents theorem, we’ve solved the problem and found the values of the two unknowns. 𝑥 is equal to 60, and 𝑦 is equal to 240.