Special Right Triangles Geometry 7-3

Special Right Triangles 45 – 45 - 90 Geometry 7-3a

Review

Areas

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

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

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

Converse of Pythagorean

New Material

Copy the following chart into your notes Investigation

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

Finish the chart for each of the listed leg lengths Investigation

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

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.

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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Special Right Triangles 30 – 60 - 90 Geometry 7-3b

Special Triangle Investigation Draw This in your notes A large equilateral triangle Special Triangle Investigation

Special Triangle Investigation Divide the triangle in half You now have a 30° – 60° – 90° triangle Special Triangle Investigation

Special Triangle Investigation Label the triangle. Special Triangle Investigation

Special Triangle Investigation Is AC = BC? Why? Yes, Definition of isosceles triangle, or equilateral Special Triangle Investigation

Special Triangle Investigation Are the two separate triangles congruent? Why? Yes, ASA Special Triangle Investigation

Special Triangle Investigation Is AD = BD? Why? Yes, CPCTC Special Triangle Investigation

Special Triangle Investigation So, AC = AB, and AD = DB; What is the relationship between AC and AD? AC = 2 x AD Special Triangle Investigation

Special Triangle Investigation So AC = 2 AD Using the Pythagorean theorem, what is the length of CD, in terms of AD? Special Triangle Investigation

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

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

Know the basic triangles Set known information equal to the corresponding part of the basic triangle Solve for the other sides

Example

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Practice

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Pages 369 – 372 2 – 8 even, 12 – 28 even, 34 – 38 even, 47, 48 Homework