In the trapezoidal ABCD, AB / / CD, ab = 2CD, m and N are the midpoint of CD and BC respectively. If the vector AB = a vector am + B vector an, then a + B =? The calculation process is given

In the trapezoidal ABCD, AB / / CD, ab = 2CD, m and N are the midpoint of CD and BC respectively. If the vector AB = a vector am + B vector an, then a + B =? The calculation process is given

The derivation process omits two words: MB = ab-am, MB = cb-cm = 2nb-cd / 2 = 2Nb - (- AB / 2) / 2 = 2Nb + AB / 4 and Nb = ab-an, so: ab-am = 2 (ab-an) + AB / 4, namely: 5ab / 4 = - am + 2An, namely: ab = - 4am / 5 + 8An / 5 = AAM + ban, namely: (a + 4 / 5) am + (B-8 / 5) an = 0, because am and an

Let AM vector = a vector and an vector = B vector. Let a and B denote AB and BC

AB+BC/2=a
BC+AB/2=b
2a-b=2AB+BC-BC-AB/2=3AB/2
AB=2(2a-b)/3
Similarly, BC = 2 (2b-a) / 3

Point m is a moving point in or on the boundary of a square with side length 2, and N is the midpoint of BC. What is the maximum value of vector an multiplied by vector am?

Establishing coordinate system with D point as origin
A(0,2),B(2,2),C(2,0),D(0,0),N(2,1)
Let m (x, y) x, y ∈ [0,2]
AN=(2,-1),AM=(x,y-2)
AN*AM=2x-y+2
When x is the largest and Y is the smallest, an * am is the largest
So x = 2, y = 0, take the maximum
AN*AM=6

In the rectangular trapezoid ABCD, if AB is parallel to CD, ∠ a = 30 °, AB + CD = m, CB + Da = n, then the area of trapezoid is?
do somebody a favour

The answer is: Mn / 6
The detailed process is as follows:
If ∠ B = 90 °, the area of right angle trapezoid is s ladder = 1 / 2 (AB + DC) BC
Translate BC to D, because ∠ a = 30 °, so ad = 2BC,
It can be seen from the question that ab + CD = m, CB + Da = n, so 2BC + BC = n
So s-ladder = 1 / 2 * (m) * n / 3 = Mn / 6