Find the coordinates of P so that P partitions the segment AB in the ratio 1:3 if A(5,8) and B(−1,4)

Answer 1

Using trig.

a^2+b^2=c^2

a=8-4=4 shortest side

b=1+5=6 mid "

16+36=52^.5=7.21=hypotenuse

1/4*7.2=1.8

c2=5.4 new side c

Trig formula for rt trgl: a/(sinα)=b/(sinβ)=c

so: a/(sinα)=7.21

4/ " = "

sinα=.555

α=33.7°

β=56.3°

γ=90°

Start over; this time with the new trgl. The angles are the same.

a2/(sinα)=5.4

"/.555=5.4

a2=3

5.4^2-3^2=20.16^.5=4.49=b2

The point is 4.5+(-1)=3.5=x

8-4+3=7=y

Answer 2

Hello Lauren! How are you?

We can solve this problem by crating a vector AB, with coordinates (-1 - 5, 4 - 8) = (-6, -4)

If P partitions AB in the ratio 1:3, it means that the vector AP = 3*PB.

Therefore we have, AB = 4*AP ; AP = (1/4)*AB = (-3/2 , -1)

Lets say P = (x_P, y_P) , if the vector AP is given by (-3/2, -1)

x_P - 5 = -3/2

x_P = 7/2

y_P - 8 = -1

y_P = 7