Say base of trapezoid B is x. Say trapezoid D is a multiple of trapezoid B by factor k. Then base trapezoid D = k*x and A(D) = k^2*A(B). The base of trapezoid C is then x + kx = x*(k+1). So trapezoid C is a multiple of B by factor k+1 and A(C)=(k+1)^2*A(B). The shaded area = A(C)-A(B)-A(D) = ((k+1)^2-1-k^2)*A(B) = 2k*A(B). Area E is the result of Area(B) first horizontally stretched by factor k and then vertically multiplied by factor 2, so A(E) = A(B)*k*2 = 2k*A(B). QED. (Sorry for my poor English... I'm Dutch...)
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Say base of trapezoid B is x.
ReplyDeleteSay trapezoid D is a multiple of trapezoid B by factor k.
Then base trapezoid D = k*x and A(D) = k^2*A(B).
The base of trapezoid C is then x + kx = x*(k+1).
So trapezoid C is a multiple of B by factor k+1 and A(C)=(k+1)^2*A(B).
The shaded area = A(C)-A(B)-A(D) = ((k+1)^2-1-k^2)*A(B) = 2k*A(B).
Area E is the result of Area(B) first horizontally stretched by factor k and then vertically multiplied by factor 2, so A(E) = A(B)*k*2 = 2k*A(B).
QED.
(Sorry for my poor English... I'm Dutch...)