Special Right Triangles 45 45 90 Geometry 7

• Area of a Triangle The area of a triangle is given by

• The Pythagorean theorem In a right triangle, the sum of the squares

• Converse of the Pythagorean theorem If the square of the length of

• 45° – 90° Triangle In a 45° – 90° triangle the hypotenuse

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

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

Label the triangle. Special Triangle Investigation

Is AC = AB? Why? Yes, Definition of isosceles triangle Special Triangle Investigation

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

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

So, AC = AB, and AD = DB; What is the relationship between AC

So AC = 2 AD Using the Pythagorean theorem, what is the length of

• 30° – 60° – 90° Triangle In a 30° – 60° –

• Know the basic triangles • Set known information equal to the corresponding

• Pages 369 – 371 • 12 – 28 even, 32 – 38

• Pages 369 – 371 • 12 – 28 even, 32 – 40

Slides: 31

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

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

• 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 C Theorem b A

• 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 C Theorem b A

Converse of Pythagorean

• 45° – 90° Triangle In a 45° – 90° triangle the hypotenuse is the square root of two times as long as each leg Theorem

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

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

Label the triangle. Special Triangle Investigation

Is AC = AB? Why? Yes, Definition of isosceles triangle Special Triangle Investigation

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

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

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

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

• 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

• 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

Practice