Question

Attached.

Don't just give me an answer, I need to know how to do this.

Answer 1

Just like the midpoint formula finds the average of the two points, these are like weighted averages.

Answer 2

i.e. to find the point one-third the way from K to M, you'll be finding the average of 3 things, but 2 of those things will be point K (to weight the average towards K.

So, multiply the coordinates of K by 2, then add the coordinates of M; finally, divide by 3.

To get one-third from M to K, do the same thing, but double M's coordinates instead to weight the average in favor of M.

Answer 3

which question?

Answer 4

Let's start with the first one.

Answer 5

do you know about congruent triangles?

the hypotenuses are all equal (that is what the problem asks for)
the angles are all equal. so all 3 triangles are congruent (by angle side angle)

that tells you the x's are all the same and the x's add up to the total x difference between the two points. same for the y's for each triangle.
does that help?

Answer 6

A little. I'm just not sure how to continue from there.

Answer 7

step 1:
I would find the total x distance between the 2 points. ditto for the y difference.

Answer 8

I assume you can do that?

Answer 9

Yes.

Answer 10

5 = x
3 = y

Answer 11

based on each triangle has the same x value, what x value does each have?

Answer 12

5/3 ?

Answer 13

yes because you know 5/3+5/3+5/3 must add up to 5
and each triangle has y=1

if you start at (-2,1) where is the next point they are asking for?

Answer 14

( -1/3, 2)

Answer 15

and the 2nd point they want?

Answer 16

(4/3, 3)

Answer 17

so that is the answer

Answer 18

notice if we add (5/3,1) to the 2nd point we end up at (3,4) which is point M. it all works out.

Answer 19

can you do question 2. It is the same idea, except you go 3/5 of the way from A to B instead of 1/3 (or 2/3) of the way.

Answer 20

Yeah, I think so.

Answer 21

Here's the weighted average method I was talking about:
The points are K(-2,1) and M(3,4).
To find a point one-third the way from K, we weight the average towards that point.
First double K -> 2K(-4,1) and add it to M
2K+M= (-1,6).
Divide by 3 -> (-1/3,2)
For two-thirds from K, double M and add it to K:
K+2M = (4,9)
Divide by 3 -> (4/3,3)

The tri-points are (-1/3,2) and (4/3,3).

This method can be generalized into formula that works for dividing any segment into any number of parts.