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Rectangular Pyramids

Is the sum of the squares of two opposite sloping edges of a rectangular based pyramid equal to the sum of the squares of the other two sloping edges?

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Rhombus in Rectangle

Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.

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Similarly So

ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.

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Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Created with GeoGebra

Triangle ABC is any right angled triangle and X is a moveable point on the hypotenuse AC. The points P and Q are the feet of the perpendiculars from X to the sides of the triangle.

Find the position of X which makes the length of PQ a minimum.
NOTES AND BACKGROUND

This dynamic image is drawn using Geogebra, free software and very easy to use. You can download your own copy of Geogebra from http://www.geogebra.org/cms/ together with a good help manual and Quickstart for beginners.