Thelengthsof the sides of the of theright trianglesare found using thePythagorean Theoremas follows;ΔABN ~ ΔTBA ~ ΔTANx = 20x = 16x = 2·√7x = 15·√5124·√33·√(15)124·√(10))√(77)2·√55·√216·316x = 5·√3, y = 10·√3, z = 10x = 3·√3, y = 6, z = 6·√3x = 6·√5, y = 12, z = 12·√5GH = 2·√(46), HK = 2·√(174)The lake is 9 kilometers longWhat is the Pythagorean Theorem?Pythagorean Theoremstates that thesquareof the length of the hypotenuse side of a right triangle is equivalent to thesumof the squares of thelengthsof the legs of the right triangle.1) The location of the angles and the congruent 90° angle and a second congruent angle indicates;ΔABN ~ ΔTBA ~ ΔTANThe missing values ofxcan be obtained using Pythagorean Theorem as follows;2) AB = √(10² + 5²) = √(125) = 5·√5² = (5 + x)² -²² = (5 + x)² -²² = 10² - x²10² + x² = (5 + x)² -²10² + x² = (5 + x)² - 125² = (5 + x)² - (10² + x²) = 10·x - 7510·x - 75 = (5·√5)² = 12510·x - 75 = 12510·x = 125 + 75 = 200x = 200/10 = 20x = 203) The hypotenuse side of the right triangle with sides 8 and 4 can be found as follows;Length of the hypotenuse = √(8² + 4²) = 4·√5Length of the leg of the larger right triangle is, length = √(8² + x²)Therefore;(x + 4)² = (8² + x²) + (4·√5)²(x + 4)² - (8² + x²) = (4·√5)²8·x - 48 = 808·x = 80 + 48 = 128x = 128/8 = 16x = 164) The leg of the larger right triangle = (12 + 2)² - x² = 14² - x²14² - x² - 12² = x² - 2²2·x² = 14² - 12² + 2² = 56x² = 56/2 = 28x = √(28) = 2·√7x = 2·√75) The length of the shorter leg of the larger right triangle can be found as follows;Length of the shorter leg = (20 + 25)²- x²x² - 25² = (20 + 25)²- x² - 20²2·x² = (20 + 25)² + 25² - 20² = 2250x² = 2250/2 = 1125x = √(1125) = 15·√5x = 15·√56) x² - 4² = (32 + 4)² - x² - 32²2·x² = (32 + 4)² + 4² - 32² = 288x² = 288/2 = 144x = √(144) = 12x = 127) Letxrepresent the length of the right tringle and lethrepresent the altitude of the right triangle(PR)² = 16² - x²16² - x² - 12² = x² - 4²2·x² = 16² - 12² + 4² = 128x² = 128/2 = 64x = √(64) = 8The length of the short leg is;x= 8Length of the longer leg, PR = √(16² - x²)PR = √(16² - 8²) = 8·√3Length of the longer leg = 8·√3The square of the altitude = 16² - x² - 12²Length of the altitude = √(16² - 64 - 12²) = 4·√38) Letxrepresent the length of the shorter leg, we get;(PR)² = 18² - x²The square of the...

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