Some(?) of the Parts

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

Pericut

Two semicircle sit on the diameter of a semicircle centre O of twice their radius. Lines through O divide the perimeter into two parts. What can you say about the lengths of these two parts?

Semi-detached

Stage: 4 Challenge Level:

Correct solutions were recieved from Aashay of Farmington High,USA, Mary of Birchwood High School, Sana, Jenny, Chris and Rosion of Madras College, St. Andrews, Andrei of School 205 Bucharest and Chen of the Chinese High School, Singapore.

If a square has an area of $40$ sq units then its side is of length $\sqrt{40} = 2 \times \sqrt{10}$.

Half the side is $\sqrt{10}$.

So radius of the circle is $\sqrt{50}$.

Applying this to the square circumscribed by the whole circle (see below)

Diameter of the circle = diagonal of the square = $2 \times \sqrt 50 = \sqrt 200$

Let side of square $= x$ units. Area of the square = $x^2$

Using Pythagoras' theorem:

$x^2 + x^2 = 200$

$x^2 = 100$

Therefore the area of the square is $100$ sq units.