In the given figure, find the values of 𝑥 and 𝑦.
Let’s look at the diagram carefully. We have a circle with centre 𝑀 and then two lines 𝐴𝐶 and 𝐴𝐵, which are both tangents to the circle, drawn from the same point 𝐴. 𝑦 represents the measure of the major arc 𝐵𝐶. 𝑥 has two meanings in this question: firstly, it’s involved in the measure of the minor arc 𝐵𝐶 — two 𝑥 degrees — and it’s also the measure of the angle formed where the two tangents meet.
In order to answer this question, we need to recall key facts about tangents intersecting outside of circles. Here is the key fact we need: when two tangents intersect outside a circle, the measure of the angle formed is half the difference of the measures of the major and minor arcs. Let’s formulate this as an equation for the circle in this question.
The measure of the angle formed is 𝑥, the measure of the major arc is 𝑦 degrees, and the measure of the minor arc is two 𝑥 degrees. So the difference is 𝑦 minus two 𝑥. We then need to half this. So we have the equation 𝑥 is equal to 𝑦 minus two 𝑥 all over two.
Now, this is one equation with two unknown letters. So we can’t solve it directly. Instead, let’s just simplify this equation first. And we’ll begin by multiplying both sides by two. This gives the equation two 𝑥 is equal to 𝑦 minus two 𝑥.
Next, I’ll simplify by grouping like terms. So I’m going to add two 𝑥 to both sides of the equation. This gives four 𝑥 is equal to 𝑦. So we don’t yet know the exact values of 𝑥 and 𝑦, but we do know a relationship that exists between them. 𝑦 is four times as big as 𝑥.
In order to determine the values of 𝑥 and 𝑦, we need a second equation that we can use. In any circle, the sum of the measures of the major and minor arcs is always equal to 360 degrees. For our circle, this means that two 𝑥, which is the measure of the minor arc, plus 𝑦, which is the measure of the major arc, is equal to 360. And so we have a second equation that we can use.
We now need to solve these equations simultaneously. As we know that 𝑦 is equal to four 𝑥, we can substitute this into the second equation to give an equation in terms of 𝑥 only. This gives two 𝑥 plus four 𝑥 is equal to 360. Simplifying the left-hand side of this equation, we have six 𝑥 is equal to 360.
In order to solve for 𝑥, we need to divide both sides by six. This gives the value of 𝑥. 𝑥 is equal to 60. Next, we need to find the value of 𝑦. And remember we have the equation 𝑦 is equal to four 𝑥. So to calculate 𝑦, we multiply 60 by four. 𝑦 is equal to 240. So we have our solution to the problem — the values of 𝑥 and 𝑦. 𝑥 is equal to 60, 𝑦 is equal to 240.
Remember the key fact we used in this question: when two tangents intersect outside a circle, the measure of the angle formed is half the difference of the measures of the major and minor arcs.