Pythagoras Theorem and Its Applications

Chapter 1 Pythagoras Theoremand Its Pythagoras Theorem and its Pythagoras TheoremThe lengthsa b<cof the sides of a right triangle satisfy the relationa2+b2= Converse TheoremIf the lengths of the sides of a triangles satisfy the relationa2+b2=c2,thenthe triangle contains a right : Euclidean a triangle withBC=a,CA=b,andAB=csatisfy-inga2+b2=c2 . Consider another triangleXY ZwithYZ=a, XZ=b,6 XZY=90 .By the Pythagorean Theorem ,XY2=a2+b2=c2,sothatXY= the triangles4 ABC 4XY Zby the SSS test. This means that6 ACB=6 XZYis a right Dissect two given squares into triangles and quadrilaterals and re-arrange the pieces into a equilateral triangles inside a linesAXandAYare extended to intersectBCandCDrespec-tively atPandQ. Show that(a)AP Qis an equilateral triangle;(b)4AP B+4 ADQ= : Euclidean a triangle with a right angle atC. If the median on the sideais the geometric mean of the sidesbandc, show thatc= (a) Supposec=a+kbfor a right triangle with legsa,b, and <k<1, anda:b:c=1 k2:2k:1+k2.

1.2 Euclid’s Proof of Pythagoras Theorem 1.2.1 Euclid’s proof C C C C B B B B A A A A 1.2.2 Application: construction of geometric mean Construction 1 Given two segments of length a<b,markthreepointsP, A, B on a line such that PA= a, PB= b,andA, B are on the same side of P. Describe a semicircle with PB as diameter, and let the ...

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