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Sagot :
In general, the formula to find the midpoint of a line segment which ends have the coordinates (x₁,y₁) and (x₂,y₂) is given by
[tex]M=(\frac{x_2+x_1}{2},\frac{y_2+y_1}{2})[/tex]Working with the formula we are given, for the x coordinate of the midpoint we have:
[tex]x=(\frac{a}{a+b})(x_2-x_1)+x_1=\frac{a(x_2-x_1)}{a+b}+\frac{(a+b)x_1}{a+b}[/tex]In order for this to be like the previous formula, we have the following equation:
[tex]\frac{a(x_2-x_1)}{a+b}+\frac{(a+b)x_1}{a+b}=\frac{x_2+x_1}{2}[/tex]From here, we see that a+b must be equal to 2, so:
[tex]\frac{a(x_2-x_1)}{2}+\frac{2x_1_{}}{2}=\frac{ax_2-ax_1}{2}+\frac{2x_1}{2}=\frac{ax_2-ax_1+2x_1}{2}=\frac{x_2+x_1}{2}[/tex]In order for this last equation to be true, a must equal 1:
[tex]\frac{x_2-x_1+2x_1}{2}=\frac{x_2+x_1}{2}[/tex]Let's verify this with the formula for the y coordinate of the midpoint:
[tex](\frac{1}{2})(y_2-y_1)+y_1=\frac{y_2-y_1}{2}+y_1=\frac{y_2-y_1}{2}+\frac{2y_1}{2}=\frac{y_2-y_1+2y_1}{2}=\frac{y_2+y_1}{2}[/tex]Since using our previous deduction leads us to the correct formula for the y coordinate of the midpoint as well, we can conclude that a=1 and a+b=2.
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