Given 𝐴 is zero, negative eight; 𝐵 is negative five, negative eight; and 𝐶 is nine, negative six, find the general equation of the straight line that passes through point 𝐴 and bisects 𝐵𝐶.
So to help us understand what’s going on here, I’ve drawn a little sketch. So we’ve got the line 𝐵𝐶. And then we’ve got the line that we’re trying to find, so the general equation of the straight line that passes through point 𝐴. So it goes through our point 𝐴. And it bisects 𝐵𝐶. And I’ve called that point 𝑀 because it’s the midpoint of 𝐵𝐶. So the first thing we want to do is find 𝑀, so find the midpoint between 𝐵 and 𝐶.
And to help us do that, we have a general rule. And that’s to find the midpoint between two points in a line, we use this. So we’ve got 𝑥 one plus 𝑥 two divided by two to find our 𝑥-coordinate and 𝑦 one plus 𝑦 two divided by two to find our 𝑦-coordinate. And this makes sense because if you add 𝑥 one and 𝑥 two, so the two points in our line either end, then divide it by two, that would be the middle of those two values. So to find the midpoint of 𝐵𝐶, which we’re gonna call 𝑀, we’re gonna substitute into our general form some values. And those values are gonna be got from the information we’ve got in the question.
So 𝐵, we’re gonna have 𝑥 one is negative five and 𝑦 one is negative eight. And 𝐶, we’re gonna have 𝑥 two is nine. And 𝑦 two is negative six. So, therefore, when we substitute in these values, we’re gonna get 𝑀. So our midpoint is equal to negative five plus nine over two for our 𝑥-coordinate and negative six plus negative eight over two for our 𝑦-coordinate. So we’re gonna calculate this. But just remembering, for the 𝑦-coordinate if we’re gonna add a negative, that’s the same as subtracting.
So, therefore, we can say that our midpoint is gonna be two, negative seven. And that’s because if you have negative five and you add nine, well, you add five to get to zero and then another four. So we’re gonna get four divided by two which is two. And then if you got negative six and then you subtract eight away, that gives us negative 14. Then divide negative 14 by two, we get negative seven. So the reason this is gonna be useful, we’ll be able to see now in our general equation of a straight line because we’re trying to find out the equation of the line.
Well, the general form for the equation of the straight line that we’re gonna use is 𝑦 minus 𝑦 one is equal to 𝑀 multiplied by 𝑥 minus 𝑥 one, where m is the slope. And it’s the small 𝑚. And that’s what we’re gonna be using now. So the reason this is useful is cause we now have two points on the line that we’re looking for because we know the points 𝐴 and 𝑀. So therefore to help us find the slope, we can use a slope equation which is gonna help us find the slope between two points.
And that formula is that the 𝑚, so our slope, is equal to 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one. So what this means is the change in 𝑦 divided by the change in 𝑥. So now, what I’ve done is I’ve got our two points 𝐴 and 𝑀 and I have labelled our coordinates 𝑥 one, 𝑦 one; 𝑥 two, 𝑦 two. So, therefore, the slope is gonna be equal to negative seven minus negative eight divided by two minus zero. Well remembering, on the numerator if we subtract a negative, then it’s the same as adding. So, therefore, our slope is gonna be equal to a half because negative seven add eight is one. One over two is a half.
Okay, great, so now we have the slope. So when we substitute what we have now into our general form of the equation of a straight line, we’re gonna have 𝑦 minus and then we’ve got negative eight, and that’s cause our 𝑦 one is negative eight, is equal to a half multiplied by 𝑥 minus zero. And that’s cause zero was our 𝑥 one. Again, reminding ourselves if you subtract a negative it turns into an add. We can rewrite our equation. So we’ve got 𝑦 plus eight is equal to a half 𝑥. And that’s because if we expand the parentheses, we have a half multiplied by 𝑥 which gives us a half 𝑥. And a half multiplied by zero is just zero.
So then to tide the equation up, we can multiply each side of the equation by two. And we do that because we want a whole 𝑥 and not a half 𝑥. And whatever you do to one side of the equation you must do to the other. So we get two multiplied by 𝑦 plus eight is equal to 𝑥. So then when we expand the parentheses, we get two 𝑦 plus 16 is equal to 𝑥. So then if we subtract 𝑥 from each side of the equation, we can say that the general equation of a straight line that passes through point 𝐴 and bisects 𝐵𝐶 is going to be negative 𝑥 plus two 𝑦 plus 16 is equal to zero.