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.

3Answer: The distance from the center to the longer chord is 13. From this, the radius of the circle is 85. More generally, if these chords has lengths 2a and 2b,andthedistance between them isd, the radius r of the circle is given by r2 = [d2 +(a− b)2][d2 +(a + b)2] 4d2.

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