Question

Pls help ya'll I have class at 4:30 (35 mins away!) and I have 3 problems left and I'm SO stuck... pls try to answer as fast as possible and with a short explanation... thanks:

The sum of the lengths of two opposite sides of the circumscribed quadrilateral is 12 cm, the length of a radius of the circle is 2 cm. Find the area of the quadrilateral.

Answer 1

The perimeter of the quadrilateral is 24 because it is a circumscribed polygon*, so the area is (24×2)/2=24cm²

* Firstly draw such a shape, and also draw the radius of the circle and the distance between the a vertex and the center (in the picture, excuse me for bad drawing).

Now consider the two triangles created: we know that they are equal because one of their sides is shared between them, the other one is a radius, and one angle is 90 degrees because it's the definition of circumscribed polygon.

So the yellow side will also be equal. We can demonstrate this for each of the four pairs of triangles, so, if we call the sides AB, BC, CD and AD, where AB is the opposite of CD, we now know for certain that AB+CD=BC+AD