Pythagoras' theorem

Can you make a hypothesis to explain these ancient numbers?

A fire-fighter needs to fill a bucket of water from the river and take it to a fire. What is the best point on the river bank for the fire-fighter to fill the bucket ?.

A ribbon is nailed down with a small amount of slack. What is the largest cube that can pass under the ribbon ?

Calculate the ratio of areas of these squares which are inscribed inside a semi-circle and a circle.

Two circles touch, what is the length of the line that is a tangent to both circles?

If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?

Shows that Pythagoras for Spherical Triangles reduces to Pythagoras's Theorem in the plane when the triangles are small relative to the radius of the sphere.

A dot starts at the point (1,0) and turns anticlockwise. Can you estimate the height of the dot after it has turned through 45 degrees? Can you calculate its height?

If two of the sides of a right-angled triangle are 5cm and 6cm long, how many possibilities are there for the length of the third side?

A rectangular plank fits neatly inside a square frame when placed diagonally. What is the length of the plank?