Pythagoras Lessons

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1 Vocabulary: 3 5 0.75 which can bewritten as aratio is called arational number and 2 which can't bewritten as ratios arecalled irrational numbers or or thesymbol is called aRADICAL n thenumber beneath the t radical sign is theRADICAND n is theprincipal or positivesquareroot of n n is thenegativesquareroot of n n isthepositiveor negativesquareroot of n ÷ ± n is thepositiveor negativesquareroot of n ± What is the approximate value of these radicals? Think about how close they are to the
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  2 Solving Equations with Radicals 2 there are actually two solutions. Can you think of them both? x4  which is written as 2 x2 or 2    The rest of the steps in solving equations just use the reverse operations that we have used before. 2 2x39   2 add 3 on both 2x12 sides  2 divide 2 on both s6sx ide  square root on both sx6 ides   2 3x152   2 add 1 on both 3x6 2sides  2 multiply by 2 on both 3x12sside  divide by 3 on both s4sx ide  square root on both sdsxie 2   2 3x154  2 multiply by 4 on bot3x12h 0essid    2 add 1 on both 3x21 sides  2 divide by 3 on both s7sx ide  suare root on both sx7 ides  2 2x45117   2 minus 5 on bot2xh 46s7side  2  72x o44n.s2 b    2  4 o2x3.s8n b   2  2 onsx19 b.   x19   2 4.2x165  x7  2 3x2452.    x10  2 3(m1)63.2  m3   2 1.5y411   y2  2 4g21035.    g3   2 5(h1)16.9    h3   2 y.273 3   y2   22 5n11382n.     n2  2 3z289.5   z10  2 53b110.2     bno solution  2 3(5z1411.2)7   z2  2 52(4t11121.)    t1   Finding the Area of Complex Shapes–see Looking For Pythagoras. Investigation 2 1. Make a rectanglearound the shape.2. Make triangles and rectangles in the outside.  A  BCDE n e area o e recane.     Find the area of the outside shapes.12B12   11A0.52   C212    22D22   31E1.52   Find the total outside area7  Subtract the area of the outside fromthe rectangle area to get the inside area.5  Find the area of the rectangle.5420    BC ADE Find the area of the outside shapes.B212    32A32   12C12   41E22   Find the total outside area14  Subtract the area of the outside fromthe rectangle area to get the inside area.6  34D62    3 Finding the Length of Segments–see Looking For Pythagoras. Page20 Make a squarearound the segment.Counting the number of dots on each side makes it easy. n e area o e ouse square.     Find the area of the outside triangles. ) 514(102   Find the area of the inside square.361026    Therefore the length of one sideof the square is26n e area o e ouse square.     Find the area of the outside triangles. ) 324(122   Find the area of the inside square.251213    Therefore the length of one sideof the square is13 Pythagorean Theorem. See page 482 in the textbook.In a right triangle the area of a square on the hypotenuse is equal to the sum of the area of the squares on the other two sides. Note: The hypotenuse is the long side of the triangle. It is opposite the right angle. 222 abc   The sum of the areas gives the equation: When the hypotenuse is the unknown. 222 abc   222 35x   34x  x34  When a leg is the unknown. 222 abc   222 5x11   222 x115   2 x96  x96  x46   4 When the side lengths include variables. 222 abc   222 (2x)(3x)8   22 4x9x64   2x83x 2 13x64  64x13  813  222 abc       222 5p67p   222 25p649p   22 624p  2  p2  3 p2  When there is more than one triangle in the diagram. 222 98y   Choose the triangle where you know two sides. Label the third side with a variable. 2 145y  Note: We don’t need to know what y is. Solve the other triangle. 222 yx13   2 22 we know y145 so 145x13     22 x13145   2 x24  x24  x26  Dot Paper Problems:Find the perimeter of this shape.Construct right triangles and then solve using Pythagoras. 22 x    22 y5229     perimeter4132925     22 z422025     Find the perimeter for these shapes. 21062512    812520    333532    3105245    21321035282     621310   5132765751     Questions Involving Recognising Pythagorean Relationships:1. Recognising right triangles:Are the following triangles right triangles?The sides of a right triangle will make the Pythagoras equation true. Try it for this triangle 697 222 679   364981   8581   Not true so not a right triangle. 152016729054 222 151620   481400   Not true so nota right triangle. 222 121620   400400  True so it isa right triangle. 162012 222 547290   81008100  True so it isa right triangle.
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