Triangle ABC has co-ordinates A (–2 ; –2 ), B (1 ; 3) and C (6 ; 0). If E is the midpoint of AB and D is the midpoint of AC, show that DE parallel toBC.
You should use the folllowing equation that relates the slopes of two parallel lines, such that:
`m_(DE) = m_(BC)`
You need to evaluate the values of the slopes of DE and BC and then you need to compare the values to decide if the lines are parallel or not.
`m_(DE) = (y_E - y_D)/(x_E - x_D)`
`m_(BC) = (y_C - y_B)/(x_C - x_B) = (0 - 3)/(6 - 1) = -3/5`
Since the problem provides the information that D,E represents the midpoints of sides AC and AB, you need to evaluate the coordinates of points D,E such that:
`x_D = (x_A + x_C)/2 => x_D = (-2+6)/2 => x_D = 2`
`y_D = (y_A + y_C)/2 => y_D = (-2+0)/2 => y_D = -1`
`x_E = (x_A + x_B)/2 => x_E = (-2 + 1)/2 => x_E = -1/2`
`y_E = (y_A + y_B)/2 => y_E = (-2+3)/2 => y_D = 1/2`
`m_(DE) = (1/2 + 1)/(-1/2 - 2) => m_(DE) = (3/2)/(-5/2)`
Reducing duplicate factors yields:
`m_(DE) = -3/5`
Comparing the values of the slopes yields:
`m_(BC) = m_(DE) = -3/5`
Hence, evaluating the values of the slopes yields that they are equal, thus the line DE is parallel to BC.