1. ## Trapezium vector problem

ABCD is a trapezium, AB and CD are parallel sides, M is the midpoint of AD. Prove that the area of the triangle BMC is half the area of the trapezium.

I let A be the origin, then the vectors of AB, AC, and AD respectively are b, c, d. So vector AM=$\displaystyle \frac{1}{2}d$
Area of the trapezium=$\displaystyle \frac{1}{2}|b\times c|+\frac{1}{2}|c\times d|$
vector CM=$\displaystyle \frac{1}{2}d-c$ vector CB=$\displaystyle b-c$
Area BMC=$\displaystyle \frac{1}{2}|(b-c)\times (\frac{1}{2}d-c)|$
My problem is I am stuck here, I think it is just some rearrangement of the vectors, but I am no good at dot and cross-products.
Thanks

2. Originally Posted by arze
ABCD is a trapezium, AB and CD are parallel sides, M is the midpoint of AD. Prove that the area of the triangle BMC is half the area of the trapezium.

I let A be the origin, then the vectors of AB, AC, and AD respectively are b, c, d. So vector AM=$\displaystyle \frac{1}{2}d$
Area of the trapezium=$\displaystyle \frac{1}{2}|b\times c|+\frac{1}{2}|c\times d|$
vector CM=$\displaystyle \frac{1}{2}d-c$ vector CB=$\displaystyle b-c$
Area BMC=$\displaystyle \frac{1}{2}|(b-c)\times (\frac{1}{2}d-c)|$
My problem is I am stuck here, I think it is just some rearrangement of the vectors, but I am no good at dot and cross-products.
Thanks
Is it necessary to use vectors? If not:

1. The area of a trapezium is calculated by:

$\displaystyle A = \frac12 \cdot (a+c) \cdot h$

2. With $\displaystyle h_a = h_c = \frac12 h$ you can calculate the areas of the triangles with the bases a or c respectively. Subtract these areas from the area of the trapezium and you'll get the area of the triangle BMC:

3. $\displaystyle A_{\Delta(BMC)}= \frac12 \cdot (a+c) \cdot h - \frac12 \cdot a \cdot h_a - \frac12 \cdot c \cdot h_c$

$\displaystyle A_{\Delta(BMC)}= \frac12 \cdot (a+c) \cdot h - \frac12 \cdot a \cdot \frac12 \cdot h - \frac12 \cdot c \cdot \frac12 \cdot h$

4. Expand and collect like terms. You'll get $\displaystyle A_{\Delta(BMC)}= \frac14 \cdot (a+c) \cdot h$ which is half of the trapezium's area.

3. Yes, I am supposed to use vectors.

4. Refer the attached image in#2. Draw MN parallel to AB.

$\displaystyle MN = \frac{a+c}{2}$

Let the vector CB be b.

Now the area of the triangle MNC = $\displaystyle \frac{1}{2}(\frac{a+c}{2}\times\frac{b}{2})$

Area of the triangle BMC = 2*MNC = $\displaystyle \frac{1}{2}(\frac{a+c}{2}\times{b})$

= $\displaystyle \frac{1}{2}(\frac{{a}\times{b}}{2} + \frac{{c}\times{b}}{2})$

= 1/2*[ares(ABC) + ares(DBC)]

So Area(BMC) = 1/2*[area(ADC) + ares(DBC)] = 1/2[areaABCD]

5. What is N?

6. Originally Posted by arze
What is N?
N is the midpoint of BC.

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