point A is at (-3,4) and point C is at (2,-6).
Find the coordinates of point b on AC such that the ratio of AB to AC is 4:5
y=-4
To find the coordinates of point B on AC such that the ratio of AB to AC is 4:5, we can use the concept of section formula.
Step 1: Calculate the coordinates of point B.
Let the coordinates of point B be (x, y).
According to the section formula, the x-coordinate of B can be calculated as:
x = (5 * xA + 4 * xC) / (5 + 4)
Substituting the x-coordinates of A, (-3), and C, (2):
x = (5 * (-3) + 4 * 2) / (5 + 4)
x = (-15 + 8) / 9
x = -7 / 9
Similarly, the y-coordinate of B can be calculated as:
y = (5 * yA + 4 * yC) / (5 + 4)
Substituting the y-coordinates of A, (4), and C, (-6)
y = (5 * 4 + 4 * (-6)) / (5 + 4)
y = (20 - 24) / 9
y = -4 / 9
Therefore, the coordinates of point B are (-7/9, -4/9).
To find the coordinates of point B on AC, we need to determine the ratio of the distances between AB and AC. Let's follow these steps:
Step 1: Calculate the distance between points A and C using the distance formula:
Distance AC = √[(x2 - x1)² + (y2 - y1)²]
= √[(2 - (-3))² + (-6 - 4)²]
= √[(2 + 3)² + (-6 - 4)²]
= √[5² + (-10)²]
= √[25 + 100]
= √125
= 5√5
Step 2: Determine the length of AB using the given ratio and the calculated length of AC:
Length AB = (4/9) * Length AC
= (4/9) * 5√5
= (20/9)√5
Step 3: Calculate the coordinates of point B by finding the point that lies on AC, such that the distance between A and B is equal to the length AB:
Using the midpoint formula:
(x, y) = ((x1 + x2)/2, (y1 + y2)/2)
Substituting the values, we know that point B lies on the line connecting A and C:
x = (-3 + 2)/2 = -1/2
y = (4 + (-6))/2 = -1
Therefore, the coordinates of point B are (-1/2, -1).
Let B(x,y) be the point you want
Make a sketch to see that B should lie in quad IV
For the x:
(x - (-3))/(2 - (-3)) = 4/5
(x+3)/5 = 4/5
5x + 15 = 20
5x = 5
x = 1
find the y in the same way
Let me know what you get