In the trapezium $MNOP$, $MP$ is the major base and $NO$ is the minor base. Knowing that the angle $P$ is $58° 15'$, the angle $OMP$ is $21° 45''$, and the diagonal $OM$ is of $6.5$ cm, calculate the area of the trapezium.
I am having some difficulty with this problem. The formula for the area is $$A=\dfrac{(B+b)h}{2}$$. I think I know how to get the value of the major base using the law of cosines and the law of sines, but what about the minor base and the height?
Any help is greatly appreciated.
Best Answer
Actually, you can't determine the area with just this information. We can determine the major base and the height, but we can't determine the minor base.
Here's how we determine the major base and the height:
Major Base: We use the law of sines to get the equation $$\frac{|MP|}{\sin(\angle MOP)}=\frac{|OM|}{\sin(\angle P)}$$ And we have everything we need there except $\angle MOP$, which we can calculate from $\angle MOP+\angle OMP+\angle P=180$
Height: We just note that $$\sin(\angle OMP)=\frac{h}{|OM|}$$
However, we cannot find the area because we can change the size of the minor base without changing any of our fixed values (which are $\angle P$, $\angle OMP$, and the length |OM|)
Here are some pictures showing this: