The verticles of triangles BAD are B(1,3) A(7,5) and D (9.-3). If E is the midpoint of segment BA and F is the midpoint of segement AD show that
a. segement EF is parallel to BD
b. Ef = 1/2BD
To show that segment EF is parallel to segment BD, we need to show that the slopes of these two segments are equal. The slope of a segment can be found using the formula:
slope = (change in y-coordinates)/(change in x-coordinates)
Let's find the slopes of segments EF and BD:
Segment EF:
Coordinates of E = Midpoint of BA => E = ((1+7)/2, (3+5)/2) = (4, 4)
Coordinates of F = Midpoint of AD => F = ((9+1)/2, (-3+3)/2) = (5, 0)
Slope of EF = (change in y-coordinates)/(change in x-coordinates) = (0-4)/(5-4) = -4/1 = -4
Segment BD:
Coordinates of B = (1, 3)
Coordinates of D = (9, -3)
Slope of BD = (change in y-coordinates)/(change in x-coordinates) = (-3-3)/(9-1) = -6/8 = -3/4
Since the slope of segment EF (-4) is equal to the slope of segment BD (-3/4), we can conclude that segment EF is parallel to segment BD.
Now, let's find the length of segment EF and compare it to half the length of segment BD to show that EF = 1/2 BD.
Length of segment EF can be found using the formula:
distance = sqrt((x2-x1)^2 + (y2-y1)^2)
Using the coordinates of E and F, we can calculate the length of EF:
Length of EF = sqrt((5-4)^2 + (0-4)^2) = sqrt(1^2 + 4^2) = sqrt(1 + 16) = sqrt(17)
Length of segment BD can be found using the formula:
Length of BD = sqrt((9-1)^2 + (-3-3)^2) = sqrt(8^2 + (-6)^2) = sqrt(64 + 36) = sqrt(100) = 10
Now, let's compare EF and 1/2 BD:
1/2 BD = 1/2 * 10 = 5
Since the length of segment EF (sqrt(17)) is equal to 1/2 the length of segment BD (5), we can conclude that EF = 1/2 BD.
Therefore, we have shown that segment EF is parallel to segment BD (part a), and EF = 1/2 BD (part b).