In the diagram below of triangle B, C, DBCD, EE is the midpoint of start overline, B, D, end overline
BD
and FF is the midpoint of start overline, C, D, end overline
CD
. If E, F, equals, minus, 9, x, plus, 66EF=−9x+66, and B, C, equals, 7, plus, 7, xBC=7+7x, what is the measure of start overline, E, F, end overline
EF
?
In triangle BCD, EE is the midpoint of BD and FF is the midpoint of CD. By the midpoint formula, the coordinates of E are (0.5B + 0.5D, 0.5B + 0.5D) = (B+D)/2 and the coordinates of F are (0.5C+0.5D, 0.5C+0.5D) = (C+D)/2.
Given E, F = -9x + 66, we can substitute the coordinates of E and F into this equation:
(B+D)/2 - (C+D)/2 = -9x + 66
(B-C)/2 = -9x + 66
B - C = -18x + 132
But we also know that B, C = 7 + 7x, so we can substitute this into the equation:
7 + 7x = -18x + 132
25x = 125
x = 5
Now that we have found the value of x, we can find the measure of EF:
B = 7 + 7(5) = 42
C = 7 + 7(5) = 42
D = (B + C)/2 = (42 + 42)/2 = 42
Therefore, the coordinates of E are (42 + 42)/2 = 42 and the coordinates of F are (42 + 42)/2 = 42. The distance EF is the absolute difference between these two coordinates, which is 42 - 42 = 0.
Therefore, the measure of line EF is 0.