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In the trapezoid ABCD, AD / / BC, the point m is the midpoint of CD. Please guess the quantitative relationship between the area of trapezoid ABCD and the area of angle ABM
Area of trapezoid ABCD = area of triangle ABM
Take the midpoint n of AB, make AE through a, perpendicular BC to e, intersect Mn to F
Because m is the midpoint of CD
So nm is the median of ABCD
So Mn / / AD / / BC, Mn = 1 / 2 (AD + BC)
So an / Nb = AF / Fe = 1
So AF = Fe = 1 / 2ae
Because AE vertical BC, Mn / / BC
So AF is the height of the triangle anm side nm and FN is the height of the triangle NBM side nm
So the area of triangle anm = 1 / 2nm * AF = 1 / 4nm * AE, the area of triangle NBM = 1 / 2nm * Fe = 1 / 4nm * AE
Because the area of triangle ABM = the area of triangle anm + the area of triangle NBM
So the area of triangle ABM = 1 / 4nm * AE + 1 / 4nm * AE = 1 / 2nm * AE
Because AE vertical BC
So the area of trapezoidal ABCD is 1 / 2 (AD + BC) * AE
Because Mn = 1 / 2 (AD + BC)
So the area of ladder ABCD = nm * AE
Because the area of triangle ABM = 1 / 2nm * AE
So the area of trapezoid ABCD is equal to the area of triangle ABM
Another proof:
The extension lines of AM and BC intersect at G
Because AD / / BC, M is the midpoint of CD
So angle dam = angle CGM, angle ADM = angle GCM, DM = MC
So the triangle dam is equal to the triangle CGM
So the area of trapezoid ABCD = the area of triangle ABG
Because the triangle dam is equal to the triangle CGM
So am = mg
So am / Ag = 1 / 2
Because the triangle ABM is the same height as the triangle ABG
So the area of triangle ABM / area of triangle ABG = am / Ag = 1 / 2
Because the area of trapezoid ABCD = the area of triangle ABG
So the area of triangle ABM / trapezoid ABCD = 1 / 2
So the area of trapezoid ABCD is equal to the area of triangle ABM
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