Finding area of a trapezoid by having points (analytic geometry)

Question

Let $ABCD$ be a trapezoid where $AB || CD$ . We have points $A , B , C ,D$:

$A: (1,2,1) $ ,$B: (3,1,3)$,$C: (-1,0,2)$ , $D:(3,-2,6)$

We want to find the area of trapezoid.

I know it is a simple high-school question but I get two different answers from two approaches.

Approach one is that we know that the area of a Quadrilateral is half the magnitude of the cross product of the diameters. for this case diameters are: $AC , BD$

$AC = (-2 , -2 ,1) , BD = (0,3,-3)$ So $AC \times BD = (3,-6 , -6)$ so $Area = 9/2 = 4.5$

For approach two, I find $AB$ and $CD$ which are parallel. Then I find $AB + CD$. After that I find $AD \times (AB+CD)$ and its magnitude and the divide it by 2. like the normal formula for finding area of trapezoid which we get the sum of the parallels and then multiply it by height.

we have:

$AB = (2,-1,2) , CD=(4,-2,4) ,AB+CD=(6,-3,6),AD=(2,-4,5) , AD \times (AB + CD) = (-9 , 18 , 18)$

And the magnitude of $(-9,18,18)$ is 27 and $27/2 =13.5 \neq 4.5$.

I also solved this question by finding the angle between two sides and it gives me the 13.5 answer at last. My question is why we get two different answers? which one is right and why the other is wrong?

Is the points given in the question valid? because I think, $\hat{AB}$ can't be equal $\hat{CD}$ but it is with the point given in the question. $\hat{AB} = \frac{\overrightarrow{AB}}{|AB|}$

Answer 1

In your second method, the orientations on the two triangles do not agree. (Look carefully at the vectors $\overrightarrow{AD}$ and $\overrightarrow{CD}$.)

Answer 2

$h^2=45-x^2$ and $h^2=18-(3-x)^2$ we get the equation $$45-x^2=18-(3-x)^2\to x=6$$ $h=3$ and $$Area_{ABCD}=\frac{3(3+6)}{2}=\frac{27}{2}$$

$$...$$

Answer 3

Plotting your quadrilateral in Geogebra reveals that ABCD is actually a self-intersecting or crossed trapezoid with area 7.5:

Assuming you meant the trapezoid BACD, both methods yield the correct area of 13.5:

$$\frac{|\mathbf{BC} \times \mathbf{AD}|}{2} = 13.5$$

$$\frac{|\mathbf{DA} \times (\mathbf{DC} + \mathbf{BA})|}{2} = 13.5$$

Here is a simple proof that the second method works:

The second method calculates the area of the triangle AED because $\mathbf{DC} + \mathbf{BA} = \mathbf{DE}$. The triangle's area is $h(a+b)/2$, which is identical to the area of the trapezoid BACD.