In the diagram, the line 𝐴𝐷 is the perpendicular bisector of the line 𝐵𝐶. Find the value of 𝑥.
So to solve this problem, what we’re gonna use is the perpendicular bisector theorem. And this tells us that if a point is on the perpendicular bisector of a line, then it’s equidistant from the ends of the segment. So if we look at that point being our vertical line in the middle of our triangle, then we can say that the distances to the left and right of this, they’re gonna be equal. So as we know that the right-hand side of the point is equal to two 𝑥 plus four. And, therefore, the left-hand side from the point must also be equal to two 𝑥 plus four.
Well, now, if we take a look at our triangle, we’ve got two right angled triangles, if we split it into two. And we’ve got one on the left-hand side. And we’ve got one on the right-hand side. And we can see they’ve both got the bottom side that’s the same because we’ve been told that in the question. Also, they share the vertical side. So this is the same. So, therefore, the diagonal sides must also be the same because our triangle is going to be congruent. And we can see that as we’ve shown here. But it also means that if we look to the bigger triangle, so both of them put together, that it would be an isosceles triangle.
So now that we’ve shown that the diagonals are equal. And we’ve done that because triangle 𝐴𝐵𝐷 is congruent to triangle 𝐴𝐶𝐷. And we showed that using side-angle-side because we had the bottom side, the right angle, and then the vertical shared side. What we can do is we can equate our diagonal lengths to each other to find 𝑥. So when we do that, we get three 𝑥 plus one equals five 𝑥 minus 12. So then we look at the side that’s got the most 𝑥s. It’s the right-hand side. So what we’re gonna do is subtract three 𝑥 from each side of our equation.
And when we do that, we get one is equal to two 𝑥 minus 12. And then what we’re gonna do is add 12 to each side of the equation, which gives us 13 is equal to two 𝑥. And then, finally, divide by two. And we get 6.5 is equal to 𝑥. So, therefore, we can say that the value of 𝑥 in our diagram is 6.5.