The endpoint of MP are M(2,1) and P(14,10). If point K partitions MP in aratio of MK:KP=3:2, what are the coordinates of K?

Question

Available response:

4.4

2.8

6.8

4.6

9.2

6.4

11.6

8.2

Answer 1

This problem can be solved extremely easily with trigonometry, but I'll just use geometry and algebra for sake of the question.

Looking at the partition ratio, we have 5 total segments. After three segments going from M towards P we arrive at K, and two segments if we're going the other direction.

We have the two points M(2,1) and P(14,10); let's find the equation of the affine line that intersects both of them.

The slope will be m=(10-1)/(14-2) = 9/12= 3/4.

Using point - slope form,

y-1 = 3/4 (x-2)

y=(3/4)x - 6/4 +1

y=(3/4)x - 1/2

Now, let's find the length of MP. We have the length formula:

L=sqrt((14-2)² + (10-1)²)

L=sqrt((12)² + (9)²)

L=sqrt(225)

L=15

Remembering that we have 5 total chunks, we take 15/5=3. 3 is the length of each chunk. 3*3=9 is the length of MK, and 3*2=6 is the length of KP.

Now that we have the length, we can similar triangles to find the coordinates.

We have a ratio of MP:MK = (14-2) : X, or 15 : 9 = 12 : X formed by the similar right triangles. For this step it is really important to sketch out the graph and draw the triangles so that you can see where the ratios come from.

5 / 3 = 12 / X(K)

X = 12 (3/5)

X = 36/5 = 7.2

For Y we do the same thing.

MP:MK = (10-1) : Y

15/9 = 9/Y

Y= 81/15 = 5.4.

To finish this problem we now add the lengths X and y we just found to the coordinates of point M.

K = (7.2+2, 5.4+1)

K = (9.2, 6.4)

Let's check our answer! Using the affine function from earlier,

y=(3/4)x - 1/2

6.4=(3/4)x-1/2

6.9*(4/3) = 9.2

Let me know if you need another example!

-Ethan