Question

Good morning. I have another math problem I may have to have checked. I am not real good at re-checking with the distance formula.

#8
Given segment AB with A(8, -4) and B(-6, -11), if P partitions AB such that the ratio of AP to AB is 2:7, find the coordinates of P.

8+2/9 (-6 - -8)
8 + 2/9 (-14)
8+ -28/9
8/1 + -28/9= 72/9 + -28/9 =44/18=3
X=3

-4 + 2/9 ( -11 - -4)
-4 + 2/9(-7)
-4/1 + -14/9= -36/9 + -14/9= -50/9= -5.5
Y=-5.5

Are my answers correct? Also, is the easiest way to double check the answers is find out the change in X and the change in Y or do you have to use the distance formula to double check your answers?

Answer 1

I don't like the way you present your solution, all I see is a bunch of numbers

I recommend this method:
First make a sketch, then place P(x,y) in a reasonable position.
Notice the ratio is AP : AB = 2:7

for the x:
(8-x)/(8-(-6)) = 2/7
(8-x)/(14) = 2/7
8-x = 28/7 = 4
-x = -4
x = 4

for the y:
(-4-y)/(-4-(-11) = 2/7
(-4-y)/7 = 2/7
-4-y = 2
y = -6
so my P is (4, -6) and your answer is incorrect

check: AP = √((8-4)^2 + (-4+6)^2) = √20 = 2√5
AB = √(14^2 + 7^2) = √245 = 7√5
AP:AB = 2√5 : 7√5 = 2:7
slope AP = (-4+6)/(8-4) = 1/2
slope AB = (-4+11)/(8+6) = 7/14 = 1/2

To fully check you must compare the slopes AND the distances

Answer 2

Yes. I see the ratio is 2:7 but don't I have to turn the ratio into a fraction? 2/9?