Question

7. A segment with endpoints I (5, 2) and J (9, 10) is divided by a point K such that IK and KJ form a 2:3 ratio. Find the y value for K.

Answer 1

use the distance formula to calculate the distance between (5,2) and (9,10), then call that distance d

then use that distance d to calculate where K would be based on the ratio 2:3

Answer 2

So for the distance formula I got 8.94427191.

Answer 3

I would leave it in the exact form sqrt(80) to make calculations easier

Answer 4

okay.

Answer 5

since IK = 2/5 d = 2/5 * sqrt(80)

we can re-use the distance formula:

2/5 *sqrt(80) = sqrt( ( x-5)^2 + (y-2)^2 )

Answer 6

doing the other side we have

3/5 * sqrt(80) = sqrt( ( 9-x)^2 + (10-y)^2 )

the algebra is kind of messy but we can treat this as a system for x and y

Answer 7

Okay. Sounds good.

Answer 8

we could square both sides of both equations to get

2/5 *sqrt(80) = sqrt( ( x-5)^2 + (y-2)^2 )
3/5 * sqrt(80) = sqrt( ( 9-x)^2 + (10-y)^2 )

(4/25)*80 = ( x-5)^2 + (y-2)^2
(9/25)*80 = ( 9-x)^2 + (10-y)^2

Answer 9

then you would just expand the binomials and eliminate (gets really messy here ;_;)

Answer 10

Great.

Answer 11

I forgot to include the choices, sorry!
4.6
5.4
4.8
5.2

Answer 12

let me try the algebra on paper

Answer 13

after doing some expansion and elimination I have x = 17 - 2y

so then you would just back-substitute to find y I believe

Answer 14

Let me try that.

Answer 15

Wait so what would I do with the x? Do I use the choices to substitute the y?

Answer 16

yeah you would just substitute "17-2y" for x

I might have made an algebra mistake somewhere because I'm not getting a real number solution

Answer 17

Can you walk me through this, because I am lost.

Answer 18

come to think of it there might be an easier way to do this

Answer 19

Really?

Answer 20

If I'm not mistaken the difference between the y-coordinates is also 2:3, meaning that, if the total distance in y is 8 (10-2) then the y-coordinate is located at (2/5)*8 = 3.2 units above the lower point, giving us a y-coordinate of 5.2

Answer 21

Oh! I get it, this way is much simpler.

Answer 22

I fanned you.

Answer 23

thanks, interesting problem