Weekly Problem 27 - 2013
Can you find the area of a parallelogram defined by two vectors?
The seven pieces in this 12 cm by 12 cm square make a Tangram set. What is the area of the shaded parallelogram?
Congratulations Christian,Nisha and Suzann, The Mount School York for seeing just how simple this problem really is - just two right angled triangles. Well done also to Sarah and Alice from the same school; they used 'half ab sin C ' but 'half base times height' is really all that is
The kite is made up of two triangles if we divide it along it's axis of symmetry.
If we call b the base of the triangle then its maximum height has got to be a , see the diagram below.
The area of this triangle is:
and h will always be smaller than a as a is the hypotenuse of the right-angled triangle made up of the green lines and side a .
The maximum value of the area of the triangle will be when h = a , so the maximum area of the triangle is :
and the area of the kite will just be b x a
If the rods are made into a parallelogram it won't change things because the parallelogram is made up of the same two triangles as the kite.
This means that the parallelogram of maximal area is actually a rectangle, well done!