Answer :
The measure of BC is 48x - 168.
In the given diagram of triangle BCD, we are told that E is the midpoint of BD and F is the midpoint of CD. Let's denote the measure of BC as y.
Since E is the midpoint of BD, we can say that BE = ED. Similarly, F is the midpoint of CD, so CF = FD.
Using the information given, we can write the equation EF = -9x + 82. Since F is the midpoint of CD, we can substitute FD for EF, so FD = -9x + 82.
Also, we know that BC = 80 - 6x.
In triangle BCD, we can write the equation BC = BE + EF + FD.
Substituting the given values, we have:
80 - 6x = (BE) + (-9x + 82) + (-9x + 82).
Simplifying the equation:
80 - 6x = BE - 9x + 82 - 9x + 82.
Combining like terms:
80 - 6x = BE - 18x + 164.
Moving the variables to one side and constants to the other side:
6x - BE + 18x = 164 - 80.
Combining like terms:
24x - BE = 84.
Rearranging the equation:
BE = 24x - 84.
Since E is the midpoint of BD, we can write:
BE = BD/2.
Substituting the value of BE:
BD/2 = 24x - 84.
Simplifying the equation:
BD = 48x - 168.
We are required to find the measure of BC, which is y. From the equation BC = BD, we have:
y = 48x - 168.
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