Given that 𝐴𝐶 equals two 𝑥 minus three centimetres, find 𝑥 and 𝑦 to the nearest thousandth.
Let’s have a look at this diagram more closely. It consists of two circles with centres 𝑁 and 𝑀 and then three lines 𝐴𝐵 and 𝐴𝐶 and 𝐴𝐷, all of which start to the common point 𝐴 outside the circles and then meet a point on their circumference. In fact, the lines 𝐴𝐵, 𝐴𝐶, and 𝐴𝐷 are all tangents. 𝐴𝐵 and 𝐴𝐶 are tangents to the circle 𝑀. And 𝐴𝐶 and 𝐴𝐷 are tangents to the circle 𝑁.
We’re asked to determine the values of 𝑥 and 𝑦, which are involved in expressions for the lengths of two of these tangents. So what we’re going to be interested in is the relationship that exists between the lengths of these tangents.
We recall then that if two tangents are drawn from the same exterior point to a circle, then they are equal in length. This means that the two tangents drawn from the point 𝐴 to circle 𝑀 are equal in length. So we have that 𝐴𝐵 is equal to 𝐴𝐶. And also the two tangents drawn from the point 𝐴 to the circle 𝑁 are equal in length. So we have that 𝐴𝐶 is equal to 𝐴𝐷. In fact, all three tangents are equal in length.
Now we can start to form some equations. We’re given in the diagram that the length of 𝐴𝐵 is 19 centimetres. And in the question, we’re told that the length of 𝐴𝐶 is two 𝑥 minus three centimetres. So substituting these values or expressions for 𝐴𝐵 and 𝐴𝐶, we have the equation 19 equals two 𝑥 minus three. We can solve this equation for 𝑥 by first adding three to each side, giving 22 is equal to two 𝑥. Then, we can divide both sides of the equation by two, giving 11 equals 𝑥. So we found the value of 𝑥. 𝑥 is equal to 11.
Now we can form our second equation, which involves 𝑦. The length of 𝐴𝐷 is 𝑦 minus five centimetres. And although the length of 𝐴𝐶 was given as two 𝑥 minus three centimetres, we know that it’s also equal to 𝐴𝐵, which is 19 centimetres. So we can form the equation 19 is equal to 𝑦 minus five. To solve for 𝑦, we just need to add five to each side of this equation, giving 24 is equal to 𝑦. We found the values of 𝑥 and 𝑦. But in the question, it asks us to give these values to the nearest thousandth. They’re actually integer values. But if we want to write them to the nearest thousandth, we’ll need to include three zeros after the decimal point.
So we have the values of 𝑥 and 𝑦. 𝑥 is equal to 11.000 and 𝑦 is equal to 24.000. Remember the key result that we used in this question was that if two tangents are drawn from the same exterior point to a circle, then they are equal in length.