Edexcel 2017, Q8) A square, with sides of length x cm, is inside a circle. Each vertex of the square is on the circumference of the circle. The area of the circle is 49cm. Work out x.

First we should draw what the question tells us.Draws circle with square inside it
We want to find out some extra information about the circle based on what we have been told.
How can we do this?Well we know what the area of the circle is, so we can find out the radius of the circle.
Using the formula: Area = πr² , 49 = πr², rearranging to give us r = √(49/π) = 3.949...
Now that we have the radius, how can we use this information to help us?
Well if we split our square in half diagonally, we can see that it gives us two equilateral triangles each with a hypotenuse length of the diameter of the circle (i.e. 2 x radius).
Using this information we can use Pythagorus' theorem.
Where the two smaller sides squared are equal to the hypotenuse squared.
So x² + x² = (2*3.949...)²
--> 2x² = 62.3887--> x² = 62.3887/2--> x = 5.6