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Special Right Triangles
Geometry 7-3
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Special Right Triangles 45 – 45 - 90
Geometry 7-3a
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Review
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Areas
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Area of a Triangle The area of a triangle is given by the formula A = ½ B x H, where A is the area, B is the length of the base, and H is the height of the triangle Area
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Theorem The Pythagorean theorem
In a right triangle, the sum of the squares of the legs of the triangle equals the square of the hypotenuse of the triangle B c a A C b Theorem
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Theorem Converse of the Pythagorean theorem
If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle. B c a A C b Theorem
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Converse of Pythagorean
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New Material
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Copy the following chart into your notes
Investigation
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Find the length of the hypotenuse of each isosceles right triangle
Find the length of the hypotenuse of each isosceles right triangle. Simplify each square root. Record the answers in your chart Investigation
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Finish the chart for each of the listed leg lengths
Investigation
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Theorem 45° – 45° – 90° Triangle
In a 45° – 45° – 90° triangle the hypotenuse is the square root of two times as long as each leg Theorem
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Question When the problem says this, How do we reduce the square root of two? Answer We don’t, unless it is in the denominator.
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Know the basic triangle
Set known information equal to the corresponding part of the basic triangle Solve for the other sides
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Example
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Practice
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Example
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Practice
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Practice
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Practice
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Practice
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Practice
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Special Right Triangles 30 – 60 - 90
Geometry 7-3b
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Special Triangle Investigation
Draw This in your notes A large equilateral triangle Special Triangle Investigation
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Special Triangle Investigation
Divide the triangle in half You now have a 30° – 60° – 90° triangle Special Triangle Investigation
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Special Triangle Investigation
Label the triangle. Special Triangle Investigation
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Special Triangle Investigation
Is AC = BC? Why? Yes, Definition of isosceles triangle, or equilateral Special Triangle Investigation
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Special Triangle Investigation
Are the two separate triangles congruent? Why? Yes, ASA Special Triangle Investigation
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Special Triangle Investigation
Is AD = BD? Why? Yes, CPCTC Special Triangle Investigation
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Special Triangle Investigation
So, AC = AB, and AD = DB; What is the relationship between AC and AD? AC = 2 x AD Special Triangle Investigation
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Special Triangle Investigation
So AC = 2 AD Using the Pythagorean theorem, what is the length of CD, in terms of AD? Special Triangle Investigation
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Theorem 30° – 60° – 90° Triangle
In a 30° – 60° – 90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is the square root of three times as long as the shorter leg Theorem
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Solving Strategy Know the basic triangles
Set known information equal to the corresponding part of the basic triangle Solve for the other sides Solving Strategy
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Know the basic triangles
Set known information equal to the corresponding part of the basic triangle Solve for the other sides
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Example
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Example
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Practice
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Practice
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Practice
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Practice
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Practice
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Practice
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Practice
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Practice
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Pages 369 – 372 2 – 8 even, 12 – 28 even, 34 – 38 even, 47, 48 Homework
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