Find the values of 𝑥 and 𝑦.
So, the first thing to note in this question is that we’re dealing with an isosceles triangle. We know that it’s an isosceles triangle because of these two lines. This is what they denote. The reason this is useful is because in an isosceles triangle we have two equal sides, which on our diagram are 𝑦 plus one and two 𝑦 minus eight, and two equal angles. And they’re the two base angles which I’ve shown here with the pink stars.
First of all, we can use the fact that we know that we’re gonna have two equal sides in our isosceles triangle to find 𝑦. Because what we can say is that 𝑦 plus one must be equal to two 𝑦 minus eight. And we’ve now formed an equation. In order to solve this equation, I want to make sure that we have the 𝑦s on one side and the numbers on the other.
And to do that, I’m going to subtract 𝑦 from each side. And when I do that, I get one is equal to 𝑦 minus eight. And that’s because if we have 𝑦 plus one and subtract 𝑦, we just get one. And if we have two 𝑦 minus eight and we subtract 𝑦 we get 𝑦 minus eight.
And next, we’re gonna add eight to each side of the equation. And we do this because we want 𝑦 on its own. And when we add eight to each side of the equation, we get nine is equal to 𝑦. So now if we just turn it the other way around, we can say 𝑦 is equal to nine and we’ve found the value of 𝑦. Okay, so, now what we can do is move on and find the value of 𝑥.
Well, the first thing we can do to help us find the value of 𝑥 is say that the measure of angle 𝐴𝐶𝐵 is equal to three 𝑥 plus six. And the reason it’s equal to three 𝑥 plus six is because it’s a base angle in an isosceles triangle. So therefore, it must be equal to the measure of the angle at 𝐴𝐵𝐶.
So, now what we can do is set up an equation that we can solve for 𝑥. And this equation is three 𝑥 plus six plus three 𝑥 plus six plus 96 is equal to 180. And the reason for this is because the angles in a triangle sum to 180 degrees. So, as you can see, I’ve given reasoning for both these sections. And the reason I’ve done that is cause whenever you’re dealing with angles, we need to give reasons for why we found certain values or equations.
So, now what we’re going to do is collect like terms. So, we’ve got three 𝑥 plus three 𝑥, which is gonna give us six 𝑥. And when we collect numbers, we’re gonna have positive six add six which is 12, add 96 which will give us 108. So therefore, we’ve got six 𝑥 plus 108 is equal to 180. So, now what we’re going to do is solve to find 𝑥.
So, what we’re gonna do is start by subtracting 108 from each side of the equation. And when we do that, we get six 𝑥 is equal to 72. And then we’re gonna divide by six because we’ve got six 𝑥 and we want to find out one 𝑥. And when we do that, we get 𝑥 is equal to 12. So therefore, we can say that we found the values of 𝑥 and 𝑦 because 𝑥 is equal to 12 and 𝑦 is equal to nine.
But what I want to do quickly, even though we’ve got the answer, is just double check our 𝑥-value. So, first of all, to enable me to do that, what I’m going to do is see what the value of three 𝑥 plus six is when I substitute in 𝑥 is equal to 12. So, when I do that I get three multiplied by 12 plus six, which is going to be 42 because three times 12 is 36. 36 add six is 42.
So, now what I’m gonna do is add 96, 42, and 42 because these should add together to make 180 if we’re correct with our value of 𝑥. And remembering we have two 42s because our base angles are the same because it’s an isosceles triangle. Well, if we add six and two, we get eight. Add another two is ten. So, we put zero in the units column and carry a one into the tens column. Then we have nine add four which is 13. Add another four which is 17. Add the one that we carried gives us 18. So, we have an eight in the tens column, a one in the hundreds column. So therefore, we’ve summed to 180, which shows us that our value of 𝑥 is correct.