Answer:
p = (4,8.5)
Explanation:
It really helps to graph these out, even if it's just a crude representation, so please see the graph I attached.
That's the points A & B plotted, with the green line connecting them.
"Divides the direct line segment from B to A in a ratio of 1:1" is just a fancy way of saying, "Cut the line in half." Or, "What's the midpoint?"
(Because a ratio of 1:1 means both pieces need to be the same length.)
So knowing all that, first just look at the graph and see about where you think the midpoint of that line segment is. Maybe somewhere around x = 3 or 4, and y = 8 or 9? Keep that in mind, because after we "do the math" we'll see if our answer makes sense.
A couple ways to do this, I'll give you the easier one. (They'll want you to use the Midpoint Formula, but it amounts to the same thing.)
See the blue lines? I drew left from A and up from B to where they intersect at (-12,15). Simple, right?
How far is it from that intersection point over to A? It's 32 units. Do you see it? 12 units to the left of the y-axis (or to the left of x=0), and 20 units to the right. 12 + 20 = 32
What's half of 32? 16, of course. Now, move left from A 16 units and where on the x-axis do you land? At 4. (Or move right 16 units from B, same answer.)
Remember when we eyeballed it and said the midpoint would be around 3 or 4 on the x-axis? Well there ya go, we just proved it's at 4.
Do the same in the y-direction:
Go up the vertical blue line from 2 until it intersects the horizontal blue line at 15. How many units is that? 13: 15 - 2
What's half of 13? 6.5.
Now go up from B 6.5 units and you get to 8.5 (2 + 6.5)
Remember when we eyeballed it and said the midpoint would be around 8 or 9 on the y-axis? Look at that: it's right between them at 8.5.
So the midpoint is at (4,8.5).
That was doing it graphically. But guess what, doing it with the Midpoint Formula is the same thing, but it's directly taking averages:
- What's the average, or middle, of the x distance? Add 20 and -12 to get 8, then divide by 2 to get 4.
- How about the y distance? Add to get 17, divide by 2 to get 8.5.
- Midpoint is (4,8.5)
When you look again at the Midpoint Formula (too hard to write here), you'll see that it's simply two averages. But what I did above was show you WHY it's like that. Hope that helps you in the future.