If A = (-4,6) and B = (6,-7), find:
(a) the coordinates of the point P on the line segment AB such that AP:Pb = 3:1
(b) the coordinates of the point P on the line AB such that AP:AB = 3:1 and P is closer to point B than to point A
The equation of AB is :
y = -13x/10 + 4/5
The equation of the perpendicular bisector of AB is: y = 10x/13 - 33/26
The distance between A and B is root(269)
The Midpoint AB is (1,-0.5)
Above Answers are all correct.
How to find the coordinates of P?
That means 3/4 of the way along the line segment from A to B
That means 3/4 in x and 3/4 in y
in x
-4 +(3/4)(6 - -4) = -4+ 3*10/4 =
= -16/4 +30/4 = 14/4 = 3 1/2
in y
6 + (3/4)(-7-6) = 6- 13*3/4
= 24/4-39/4 = -15/4 = -3 3/4
so
(3.5 , -3.75)
In part B same deal but times 3, not 3/4
Thanks a lot Damon
You are welcome
To find the coordinates of point P on the line segment AB, we can use the concept of section formula. The section formula states that if a point P divides a line segment AB in the ratio m:n, then the coordinates of point P can be found using the formula:
P = ((n * A) + (m * B)) / (m + n)
Now let's solve the given questions:
(a) Finding the coordinates of point P on the line segment AB such that AP:Pb = 3:1
Given: A = (-4,6), B = (6,-7), AP:Pb = 3:1
Using the section formula, we can substitute the values as follows:
P = ((1 * (-4,6)) + (3 * (6,-7))) / (3 + 1)
P = ((-4,6) + (18,-21)) / 4
P = (14,-15) / 4
P = (14/4, -15/4)
P = (3.5, -3.75)
Therefore, the coordinates of point P on the line segment AB such that AP:Pb = 3:1 are (3.5, -3.75).
(b) Finding the coordinates of point P on the line AB such that AP:AB = 3:1 and P is closer to point B than to point A
Given: A = (-4,6), B = (6,-7), AP:AB = 3:1
We want P to be closer to point B than to point A. This means that the ratio of AP to AB should be less than 1/2.
By substituting the values into the formula, we get:
AP:AB = 3:1
AP/(AP + AB) = 3/4
Let the ratio AP/(AP + AB) be represented by r, then we have:
r = 3/4
We can rewrite this equation as:
AP/(AP + AB) = r
AP = r * (AP + AB)
AP = r * AP + r * AB
Rearranging the equation, we get:
AP - r * AP = r * AB
(1 - r) * AP = r * AB
AP = (r * AB) / (1 - r)
Substituting the values, we get:
AP = (3/4 * (6,-7)) / (1 - 3/4)
AP = (18/4, -21/4) / (1/4)
AP = (4.5, -5.25) / (0.25)
AP = (18, -21) / 1
AP = (18, -21)
Therefore, the coordinates of point P on the line AB such that AP:AB = 3:1 and P is closer to point B than to point A are (18,-21).