In the figure below, 𝐴𝐵 is a tangent to the circle 𝑀 at 𝐵, 𝐶𝐷 is a diameter, the measure of angle 𝐵𝐴𝑀 equals 𝑥, and the measure of angle 𝑀𝐷𝐵 equals two 𝑥 minus 55. Find the value of 𝑥 in degrees.
Whenever you attempt this kind of question, I always say mark on diagram the angles that you already know or that you can work out first. Well, the first angle we know is that the angle 𝐷𝐵𝑀 is gonna be equal to two 𝑥 minus 55. And we know that it equals two 𝑥 minus 55 because it’s gonna be the same as angle 𝑀𝐷𝐵 because it’s an isosceles triangle. Well, how do we know that it’s an isosceles triangle? Well, it’s gotta be an isosceles triangle because 𝐷𝑀 will have to equal 𝑀𝐵 because they’re both radii of the circle.
Okay, great, so now what else do we know? Well, the next angle we know is angle 𝐴𝐵𝑀. We know that angle 𝐴𝐵𝑀 is equal to 90 degrees. So it’s a right angle and this is because it’s the angle between a tangent and a radius and the angle between tangent and radius is always 90 degrees. Okay, so now, we’ve actually written all the angles that we know.
So now, it’s actually time to actually calculate some additional angles. The first one we’re gonna start with is angle 𝐷𝑀𝐵. Well, we know that angle 𝐷𝑀𝐵 is gonna be equal to 180 minus two 𝑥 minus 55 minus two 𝑥 minus 55, which is gonna be equal to 180 plus 110 minus four 𝑥. Now, I’ll point out here a common mistake. The common mistake is actually to have had minus 110. And the reason you might have that is because obviously we could see that it was minus 55 minus 55. However, as it’s minus a minus, that makes it positive. So we get add 110. And we can say that angle 𝐷𝑀𝐵 is equal to 290 minus four 𝑥. And this is because the angles in a triangle add up to 180.
So now, we can move on to angle 𝐵𝑀𝐴, which is going to be equal to 180 minus then we’ve got 290 minus four 𝑥. And that’s because that was the angle 𝐷𝑀𝐵. And this gives us four 𝑥 minus 110. Again, be very careful here of the negative numbers cause we have got minus a negative, again which gives us positive four 𝑥. And again, we include our reasoning. And the reason for this is because the angles on a straight line equal 180 degrees. So therefore, angle 𝐷𝑀𝐵 and angle 𝐵𝑀𝐴 must be equal to 180 degrees.
Okay, great, so we’ve now found that angle. So now, we can actually move on and find the angle we’re looking for, which is 𝐵𝐴𝑀. So therefore, we could say that the angle 𝐵𝐴𝑀 is equal to 180 minus 90 minus four 𝑥 minus 110. Well, therefore, we can say that 𝑥 is equal to 200 minus four 𝑥. And we can say 𝑥 because we can see that angle 𝐵𝐴𝑀 is equal to 𝑥. So now, we’ve got 𝑥 is equal to 200 minus four 𝑥 and we can solve to find 𝑥. So then if we add four 𝑥 to each side, we get five 𝑥 equals 200. And then we divide both sides by five. We get 𝑥 equals 40. So therefore, we can say that the measure of angle 𝐵𝐴𝑀 is equal to 40 degrees. And therefore, 𝑥 is equal to 40 degrees.
Okay, great, so we’ve got to our final answer. But what we want to do here is actually quickly double check it. I’m going to check it by using the first triangle, which is triangle 𝐵𝐷𝑀. So we have all three angles which are two 𝑥 minus 55 plus two 𝑥 minus 55 plus 290 minus four 𝑥. So now, if I actually substitute in our value for 𝑥 which is 40, we’re gonna get 80 minus 55 because two multiplied by 40 gives us 80 plus another 80 minus 55 minus 290 minus 160 because again four multiplied by 40 equals 160.
Okay, great, so let’s calculate this. Well, this gives us 180. Well, we know that this is correct because the angles in a triangle add up to 180. So therefore, yes, we can say with confidence that 𝑥 is equal to 40 degrees.