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How many right-angled triangles are there with sides that are all integers less than 100 units?

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A tennis ball is served from directly above the baseline (assume the ball travels in a straight line). What is the minimum height that the ball can be hit at to ensure it lands in the service area?

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Weighty Problem

Age 11 to 14 Challenge Level:

The diagram shows a very heavy kitchen cabinet. It cannot be lifted but it can be pivoted around a corner. That way the cabinet can be moved in a series of turns about one corner at a time.

Cabinet on Grid

Presently it is placed (on the X) facing in the direction of the arrows.

The task is to move it, without sliding, in a series of turns about the corners so that it is facing the other way round (and still placed with its edge on the X). The corners of the cabinet do not always have to stand on the corners of the square tiles on the floor.