For which values of 𝑥 and 𝑦 is 𝐴𝐷 a perpendicular bisector of 𝐵𝐶?
We’ve been given a diagram of the line segment 𝐵𝐶 and the line 𝐴𝐷 and also the lines connecting the endpoints of 𝐵𝐶 to the point 𝐴. We’ve been given expressions for the length of each line in terms of the variables 𝑥 and 𝑦. And we’re asked to determine the values of 𝑥 and 𝑦 that will make 𝐴𝐷 a perpendicular bisector of 𝐵𝐶. Let’s consider how to approach this problem. If 𝐴𝐷 is to be a perpendicular bisector of 𝐵𝐶, then this means that the line segments 𝐵𝐷 and 𝐷𝐶 must be equal in length. We can therefore set the expressions for 𝐵𝐷 and 𝐷𝐶 equal to one another, giving us the equation five 𝑦 minus one is equal to 10 minus three 𝑥.
I can manipulate this equation slightly by first adding one to both sides and then adding three 𝑥 to both sides, giving the equation five 𝑦 plus three 𝑥 is equal to 11. Now, this is just one equation with two unknown variables, 𝑥 and 𝑦. And so we can’t solve it. We need another equation. Let’s think about this point 𝐴 which clearly lies on the line 𝐴𝐷 which we want to be a perpendicular bisector of 𝐵𝐶.
At this point, we need to recall the converse of the perpendicular bisector theorem which tells us that if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. 𝐵 and 𝐶 at the end points of the segment, so what this means is that if 𝐴 is the same distance from 𝐵 as it is from 𝐶, then 𝐴𝐷 will be the perpendicular bisector of 𝐵𝐶. We have expressions for 𝐴𝐵 and 𝐴𝐶 in terms of the variables 𝑥 and 𝑦. So we can form a second equation. Three 𝑥 plus two is equal to five 𝑦 plus three.
I can also manipulate this equation by first subtracting two from both sides and then subtracting five 𝑦 from both sides, to give the equation three 𝑥 minus five 𝑦 is equal to one. I now have two equations in two unknowns. Which means, I can solve the equation simultaneously in order to find the values of 𝑥 and 𝑦. If we look at the equations closely, we’ll see that one has five 𝑦 and the other has negative five 𝑦. Which means, if we add the two equations together, the 𝑦 terms will be eliminated. Adding equations one and two together gives three 𝑥 plus three 𝑥 which is six 𝑥 is equal to 11 plus one which is 12.
To solve this equation for 𝑥, we just need to divide both sides by six, giving 𝑥 is equal to two. Now that we know the value of 𝑥, we can find the value of 𝑦 by substituting into either of the two equations. I’m going to choose to substitute into equation one, giving five 𝑦 plus three multiplied by two is equal to 11. This simplifies to five 𝑦 plus six is equal to 11. And subtracting six from each side, we then have that five 𝑦 is equal to five. Dividing both sides of the equation by five gives the value of 𝑦. It’s equal to one. So we have that 𝑥 is equal to two and 𝑦 is equal to one.
By the converse of the perpendicular bisector theorem, we know that when 𝑥 and 𝑦 take these values, the point 𝐴 will be equidistant from the points 𝐵 and 𝐶. And therefore, the line 𝐴𝐷 will be a perpendicular bisector of 𝐵𝐶.