Rectangular Pyramids

574 /www/nrich/html/content/98/05/six5/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <table width="100%"> <tr> <td>Is it true that p 2 + q 2 = s 2 + r 2 <ol type="a"> <li>in 2D, where the rectangle ABCD and the point V are in the same plane?</li> <li>in 3D, where the diagram represents a pyramid, with vertex V, on a rectangular base ABCD?</li> </ol> </td> <td><mdo:image src="pyramid.gif" alt=""></mdo:image></td> </tr> </table> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> A proof of this result came from students from the Key Stage 3 Maths Club at Strabane Grammar School, Northern Ireland and the following one from Joel, ACS (Barker) Singapore: <mdo:image alt="" src="pyramid_soln.gif"></mdo:image> (a) In the diagram given (in 2 dimensions) p 2 + q 2 = r 2 + s 2 Proof Draw 2 lines through V parallel to the edges of the rectangle, dividing the figure into four pairs of right-angled triangles. Label the point on AB as E, on BC as F, on CD as G and on DA as H. By Pythagoras theorem: <table> <tbody> <tr> <td>p 2 = DG 2 +DH 2</td> </tr> <tr> <td>q 2 = BE 2 + BF 2</td> </tr> <tr> <td>r 2 = CF 2 + CG 2</td> </tr> <tr> <td>s 2 = AE 2 + AH 2</td> </tr> </tbody> </table> Since DG=AE, CG=BE, AH=BF and CF=DH, p 2 + q 2 = AE 2 + CF 2 + CG 2 + AH 2 = r 2 + s 2 (b) If the diagram represents a pyramid on a rectangular base where p, q, r and s are the lengths of the sloping edges then the result p 2 + q 2 = r 2 + s 2 still holds true. Proof Let V 1 be the foot of the perpendicular from V to the base ABCD of the pyramid and let h be the height of the pyramid so that VV 1 = h and let V 1 A = s 1 , V 1 B = q 1 , V 1 C = r 1 , and V 1 D= p 1 . By Pythagoras theorem we have: p 1 2 + h 2 = p 2 , q 1 2 + h 2 = q 2 , r 1 2 + h 2 = r 2 and s 1 2 + h 2 = s 2 . Using the result already proved in 2dimensions, that is p 1 2 + q 1 2 = r 1 2 + s 1 2 , we get p 1 2 + q 1 2 + 2h 2 = r 1 2 + s 1 2 + 2h 2 so p 2 + q 2 = r 2 + s 2 . </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <h3>Why do this problem?</h3> It provides experience of generalising a result from 2 dimensions to an equivalent result in 3 dimensions. This problem asks the question for them but learners should be encouraged to ask themselves "What if..." and always to think about possible generalisations. <h3>Key questions</h3> What comes to mind when a problem involves squares of distances? If we are looking for Pythagoras theorem where are the right angles triangles? <h3>Possible extension</h3> The problem <a href="http://nrich.maths.org/public/viewer.php?obj_id=1949&part=index"> Pythagoras for a Tetrahedron.</a> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Think about right-angled triangles.</mdoxml> 2 3 0 0 0 1 1 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? 2D Geometry, Shape and Space Pythagoras' theorem 3D Geometry, Shape and Space Pyramids 2D Geometry, Shape and Space Rectangles