4.11.2 Special Right Triangles
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1) The document discusses special right triangles, specifically 45-45-90 triangles and 30-60-90 triangles. 2) For 45-45-90 triangles, the hypotenuse is √2 times the length of either leg. For 30-60-90 triangles, the hypotenuse is 2 times the shorter leg and √3 times the longer leg. 3) Examples are provided to demonstrate using the properties of these special right triangles to calculate missing side lengths.
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Gch5 l8
This document is from a Holt Geometry textbook and covers applying special right triangles. It includes examples of using 45-45-90 and 30-60-90 triangles to find unknown side lengths. Key relationships covered are that for a 45-45-90 triangle the hypotenuse is √2 times the leg and for a 30-60-90 triangle the hypotenuse is 2 times the shorter leg. Examples include finding side lengths, checking solutions, and solving applied problems involving diagonal lengths of squares and equilateral triangles. The lesson ends with a quiz to test understanding of using these special right triangle relationships.
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maths jee formulas.pdf
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6.7 Exponential and Logarithmic Models
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4.11.2 Special Right Triangles
- 1. Special Right Triangles The student is able to (I can): • Identify when a triangle is a 45°-45°-90° or 30°-60°-90° triangle • Use special right triangle relationships to solve problems
- 2. Consider the following triangle: To find x, we would use a2 + b2 = c2, which gives us: What would x be if each leg was 2? 1 1 x 2 2 2 2 1 1 1 1 2 2 x x x + = = + = =
- 3. Again, we will use the Pythagorean Theorem Simplifying the radical, we can factor to give us Do you notice a pattern? 2 2 x 2 2 2 2 2 2 4 4 8 8 x x x + = = + = = 8 2 2.
- 4. 45°-45°-90° Triangle Theorem In a 45°-45°-90° triangle, both legs are congruent, and the length of the hypotenuse is times the length of the leg. 2 x x x 245° 45°
- 5. Examples Find the value of x. Give your answer in simplest radical form. 1. 2. 3. 45° x 8 x 7 9 2x (square)
- 6. Examples Find the value of x. Give your answer in simplest radical form. 1. 2. 3. 45° x 88 2 x 7 7 2 9 2x9 2 9 2 = (square)
- 7. If we know the hypotenuse and need to find the leg of a 45- 45-90 triangle, we have to divide by . This means we will have to rationalize the denominator, which means to multiply the top and bottom by the radical. The shortcut for this is to divide the hypotenuse by 2 and then multiply by 2 16 x 16 16 2 2 2 2 = = x 16 2 8 2 2 = = 2. 16 2 8 2 2 = =x
- 8. Examples Find the value of x. 1. 2. x 45° 20 x 5
- 9. Examples Find the value of x. 1. 2. x 45° 20 20 2 10 2 2 x = = x 5 5 2 2 x =
- 10. 30°-60°-90° Triangle Theorem In a 30°-60°-90° triangle, the length of the hypotenuse is 2222 times the length of the shorter leg, and the length of the longer leg is times the length of the shorter leg. Remember: the shorter leg is always opposite the smallest (30°) angle; the longer leg is always opposite the 60° angle. 3 x 2x3x 60° 30°
- 11. Examples Find the value of x. Simplify radicals. 1. 2. 3. 4. 7 x 60° 30° 11x 9 x 60° 1616 60° x
- 12. Examples Find the value of x. Simplify radicals. 1. 2. 3. 4. 7 x 60° 30° 11x 9 x 60° 1616 60° x 9 3 16 3 8 3 2 = 14 11 5.5 2 =
- 13. Examples To find the shorter leg from the longer leg: Find the value of x 1. 2. 9 x 60° 10 x 30° longer leg 3 longer leg 3 33 3 =
- 14. Examples To find the shorter leg from the longer leg: Find the value of x 1. 2. 9 x 60° 10 x 30° longer leg 3 longer leg 3 33 3 = 9 3 3 3 3 = =x 10 3 3 x =