Point A (-2,3) and point B (4,6) are the endpoints of segment AB. What are the coordinates of point C on segment AB that AC is 2/3 the length of segment AB from point A?

Please help, thanks.

Nope

To find the coordinates of point C on segment AB that creates a segment AC that is 2/3 the length of segment AB from point A, we can follow these steps:

Step 1: Find the length of segment AB.

The length of a segment AB can be found using the distance formula:

d = √((x2 - x1)^2 + (y2 - y1)^2)

For point A (-2,3) and point B (4,6), we have:

dAB = √((4 - (-2))^2 + (6 - 3)^2)
= √((6)^2 + (3)^2)
= √(36 + 9)
= √45
= 3√5

So, the length of segment AB is 3√5.

Step 2: Find the length of segment AC.

Since we want segment AC to be 2/3 the length of segment AB, we can calculate:

dAC = (2/3) * dAB
= (2/3) * 3√5
= 2√5

Step 3: Find the coordinates of point C.

To find the coordinates C (x,y) on segment AB, we need to find its position between points A and B.

We can use the idea of midpoint formula, which states that the midpoint of a line segment is the average of the x-coordinates and the average of the y-coordinates of the endpoints.

For point A (-2,3) and point C (x,y), we have:

x-coordinate of midpoint = (x1 + x2)/2
= (-2 + x)/2
= (-2 + x)/2

y-coordinate of midpoint = (y1 + y2)/2
= (3 + y)/2
= (3 + y)/2

Since segment AC is 2/3 the length of segment AB from point A, we can set up the following proportion:

dAC / dAB = 2/3

(√((x - (-2))^2 + (y - 3)^2)) / (3√5) = 2/3

(√((x + 2)^2 + (y - 3)^2)) / (3√5) = 2/3 [Since (-a)^2 = a^2]

Solving this equation will give us the coordinates of point C. However, it is a complex computational problem, and the exact values of x and y coordinates cannot be given without knowing the specific values of x and y.

To find the coordinates of point C, we need to determine the position of a point that is 2/3 the distance from point A towards point B on segment AB.

First, let's calculate the length of segment AB. The distance between two points (x1, y1) and (x2, y2) can be found using the distance formula:

d = √((x2 - x1)^2 + (y2 - y1)^2)

For segment AB, the coordinates are A(-2, 3) and B(4, 6):

d_AB = √((4 - (-2))^2 + (6 - 3)^2)
= √(6^2 + 3^2)
= √(36 + 9)
= √45
= 3√5

Now, to find point C that is 2/3 the distance from A towards B, we need to multiply the length of segment AB by 2/3:

d_AC = (2/3) * d_AB
= (2/3) * 3√5
= 2√5

We can interpret this as moving 2√5 units from point A towards point B.

To find the x-coordinate of point C, we add 2√5 to the x-coordinate of point A:

x_C = x_A + 2√5
= -2 + 2√5

To find the y-coordinate of point C, we add 2√5 to the y-coordinate of point A:

y_C = y_A + 2√5
= 3 + 2√5

Therefore, the coordinates of point C on segment AB, where AC is 2/3 the length of segment AB from point A, are:

C(-2 + 2√5, 3 + 2√5)

C is 23 of the way from A to B

A(-2,3) + (6,3) + = B(4,6)

So, add 2/3 of (6,3), or (4,2) to A.

A(-2,3) + (4,2) = C(2,5)