A line segment AE with coordinates A(6,-6) and E (14,2)is decided into four equal segments by BC and D. Calculate the coordinates of C,B,D
E-A = (8,8)
So each segment has length (2,2)
B = A+(2,2) = (8.-4)
and similarly for C = B+(2,2), D = C+(2,2)
Helpful but I really don't understand the process
To calculate the coordinates of the points C, B, and D, we need to find the midpoint of the line segment AE, which we can then use to find the coordinates of each of these points.
Step 1: Find the coordinates of the midpoint of AE.
The midpoint of a line segment with endpoints (x1,y1) and (x2,y2) can be calculated using the following formulas:
Midpoint_x = (x1 + x2) / 2
Midpoint_y = (y1 + y2) / 2
In this case, the coordinates of A are (6,-6) and the coordinates of E are (14,2). Plug these values into the midpoint formulas:
Midpoint_x = (6 + 14) / 2 = 20 / 2 = 10
Midpoint_y = (-6 + 2) / 2 = -4 / 2 = -2
So, the coordinates of the midpoint of AE are (10, -2).
Step 2: Find the coordinates of C, B, and D.
Since we want to divide the line segment AE into four equal segments, the distance between A and C should be one-fourth of the distance between A and E. To find the coordinates of C, we move one-fourth of the distance from A toward the midpoint.
First, find the distance between A and E using the distance formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Distance = √((14 - 6)^2 + (2 - (-6))^2)
Distance = √(8^2 + 8^2)
Distance = √(64 + 64)
Distance = √128
Distance ≈ 11.31
Since we want to divide the distance AE into four equal segments, each segment will have a length of one-fourth of the distance:
Segment_length = Distance / 4
Segment_length ≈ 11.31 / 4
Segment_length ≈ 2.83
Now, to find the coordinates of C, move one-fourth of the distance from A toward the midpoint:
C_x = A_x + (Segment_length * (Midpoint_x - A_x) / Distance)
C_y = A_y + (Segment_length * (Midpoint_y - A_y) / Distance)
C_x = 6 + (2.83 * (10 - 6) / 11.31)
C_y = -6 + (2.83 * (-2 - (-6)) / 11.31)
Simplifying the equations:
C_x = 6 + (2.83 * 4 / 11.31) ≈ 7.97
C_y = -6 + (2.83 * 4 / 11.31) ≈ -4.8
Therefore, the approximate coordinates of point C are (7.97, -4.8).
To find the coordinates of point D, we continue moving one-fourth of the distance from C towards the midpoint:
D_x = C_x + (Segment_length * (Midpoint_x - C_x) / Distance)
D_y = C_y + (Segment_length * (Midpoint_y - C_y) / Distance)
D_x = 7.97 + (2.83 * (10 - 7.97) / 11.31)
D_y = -4.8 + (2.83 * (-2 - (-4.8)) / 11.31)
Simplifying the equations:
D_x = 7.97 + (2.83 * 2.03 / 11.31) ≈ 8.39
D_y = -4.8 + (2.83 * 2.8 / 11.31) ≈ -3.83
Therefore, the approximate coordinates of point D are (8.39, -3.83).
Finally, to find the coordinates of point B, we move one-fourth of the distance from A towards C:
B_x = A_x + (Segment_length * (C_x - A_x) / Distance)
B_y = A_y + (Segment_length * (C_y - A_y) / Distance)
B_x = 6 + (2.83 * (7.97 - 6) / 11.31)
B_y = -6 + (2.83 * (-4.8 - (-6)) / 11.31)
Simplifying the equations:
B_x = 6 + (2.83 * 1.97 / 11.31) ≈ 6.51
B_y = -6 + (2.83 * 1.2 / 11.31) ≈ -5.22
Therefore, the approximate coordinates of point B are (6.51, -5.22).
To summarize:
Coordinates of C ≈ (7.97, -4.8)
Coordinates of B ≈ (6.51, -5.22)
Coordinates of D ≈ (8.39, -3.83)