Question

The coordinates of the endpoints of CD are C(-7,9) and D(8,-6). Point E is on CD and divides it such that CE:DE is 2:1. Are the coordinates of E?

a) (-5, 7)
b) (-3, 5)
c) (-1, 3)
d) (1, 1)

Answer 1

The coordinates of point E on CD, dividing it in a 2:1 ratio, are (-5, 7).

Step-by-step explanation:

To find the coordinates of point E, we need to find the midpoint of CD and then divide that segment into a 2:1 ratio. First, we find the midpoint of CD by averaging the x-coordinates and the y-coordinates. The midpoint is M((-7 + 8)/2, (9 + (-6))/2) = (1/2, 3/2). Now, we use the midpoint formula to find the coordinates of point E. Let E(x, y) be the coordinates of point E. We know that CE:DE is 2:1, so the distance between C and E is twice the distance between D and E. Using the distance formula, we can set up two equations:

(x - 1/2)^2 + (y - 3/2)^2 = 4[(x - 8)^2 + (y + 6)^2]

2(x - 1/2)^2 + 2(y - 3/2)^2 = (x - 8)^2 + (y + 6)^2

Expanding and simplifying the equations gives:

5x^2 - 45x + 81 + 5y^2 + 57y + 261 = 0

x^2 + y^2 - 9x - 11.4y - 36 = 0

This equation represents a circle. By solving for x and y, we get x = -5 and y = 7. Therefore, the coordinates of point E are (-5, 7).