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 2:3. What are the coordinates of e? a) (0, 0) b) (1, 1) c) (-3, 3) d) (4, -4)

Answer 1

The coordinates of point E, which divides segment CD with endpoints at C(-7, 9) and D(8, -6) into a ratio of 2:3, are calculated using the section formula and found to be (-1, 3), corresponding to option c).

Step-by-step explanation:

The student is asking how to find the coordinates of point E which divides the segment CD into a ratio of 2:3, where C is at (-7, 9) and D is at (8, -6). To solve for the coordinates of E, we can apply the section formula. The section formula provides a way to find the coordinates of a point that divides a line segment internally in a given ratio (m:n). The formula for the x-coordinate is given by (mx2 + nx1)/(m + n), and the formula for the y-coordinate is (my2 + ny1)/(m + n). We plug in the coordinates of C and D and the ratio values (2 for CE and 3 for DE) into the formula:

• x-coordinate of E = (2*8 + 3*(-7))/(2 + 3) = (16 - 21)/5 = -5/5 = -1
• y-coordinate of E = (2*(-6) + 3*9)/(2 + 3) = (-12 + 27)/5 = 15/5 = 3

Therefore, the coordinates of point E are (-1, 3), which matches option c).

Answer 2

The coordinates of point e are (-1, 0).

Step-by-step explanation:

Point e divides the line segment cd into two parts, ce and de, in the ratio 2:3. To find the coordinates of e, we can use the section formula. Let (x, y) be the coordinates of e. The formula for finding the coordinates of a point dividing a line segment in the ratio m:n is given by:

`\[x = (mx_2 + nx_1)/(m + n), \quad y = (my_2 + ny_1)/(m + n)\]`

In this case, the coordinates of c are (-7, 9) and the coordinates of d are (8, -6). The ratio of ce to de is 2:3. Substituting these values into the formula, we get:

`\[x = (2(8) + 3(-7))/(2 + 3) = -1\]`

`\[y = (2(-6) + 3(9))/(2 + 3) = 0\]`

Therefore, the coordinates of e are (-1, 0). This means that option c) (-3, 3) is not the correct answer. The correct answer is option a) (-1, 0), as calculated using the section formula for dividing a line segment in a given ratio.