The Pythagorean Theorem & Special Right Triangles

Transcription

1 Theorem 7.1 Chapter 7: Right Triangles & Trigonometry Sections 1 4 Name Geometry Notes The Pythagorean Theorem & Special Right Triangles We are all familiar with the Pythagorean Theorem and now we ve explored one proof there are 370 known proofs, by the way! let s put it in to practice. Pythagorean Theorem In a triangle, the of the length of the is equal to the of the of the lengths of the. Refresh your memory: the hypotenuse is And, the legs are Radical, dude. Since we will be dealing with square, we want to also refresh our skills in this area. First: does every number have a square root?, but remember! It may not be a number. Estimates versus simplified radicals You should be pretty good at problems like these: 36 = OR 49 = OR 121 = How do you solve something like this without a calculator? 28 Let s break it down: This answer,, is actually the answer, whereas what you get from a calculator,, is only an.

2 Try some more: Estimate: Estimate: Estimate: A little more radical practice 2 3x x x 70 How about this one? 5 3 you can t leave a radical in the denominator Page 2 of 12

3 Try this: AB = 8, BC = 6, AC =? Can you think of 3 other side lengths that will come out to be perfect like the one above? (Hint: look at the side lengths above and see if they are multiples or factors of similar numbers.) These special sets of positive are called Triples and can be used to make angles where there are none. Page 436 7: 3 5, and more. Practice 7.1 Page 3 of 12

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5 The Converse of the Pythagorean Theorem Up to now, we ve said the Pythagorean Theorem, + =, is used only with right triangles. As you might have suspected, a version of it can also be used with all other triangles. Try it! Complete the chart below by finding values for c that make the equation/inequality true. The catch! c must be greater than either a or b, but less than a + b. Equation / Inequality a = b = a + b c =? (> a, > b, < a + b) a 2 + b 2 = c a 2 + b 2 < c a 2 + b 2 > c a 2 + b 2 = c a 2 + b 2 < c a 2 + b 2 > c a 2 + b 2 = c a 2 + b 2 < c a 2 + b 2 > c Construct these triangles; you may use Patty Paper or simply draw them on scrap / white paper. 3. Make a conjecture about the type of triangle that results for each of the following possibilities: a. a 2 + b 2 = c 2 b. a 2 + b 2 < c 2 c. a 2 + b 2 > c 2 Page 5 of 12

6 Theorems 7.3 & 7.4 Theorem 7.2 Chapter 7: Trigonometry... Sections 1 4 Converse of the Pythagorean Theorem If the sum of the of the lengths of two sides of a triangle the square of the length of the third side, then the triangle is a triangle and the longest side is the. Pythagorean Inequality Theorems If the sum of the of the lengths of the two sides of a triangle is than the square of the length of the longest side, then the triangle is. If the sum of the squares of the lengths of the shorter sides of a triangle is than the square of the length of the side, then the triangle is. Page 444: 3 5, and more. Practice 7.2 Page 6 of 12

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8 Special Right Triangles What do you get when you cut a square in half? An triangle, also called a because of its angle measurements. Why so special? Complete the chart. Let s do some calculating: find the length of the hypotenuse of the isosceles right triangle using the given values. Keep your answer in the simplest radical form. Leg (a) Leg (b) Hypotenuse (c) 3 a b c x Work area Notice anything? Page 8 of 12

9 What do you get when you cut an equilateral triangle in half? An triangle, also called a because of its angle measurements. Why so special? Let s start with some deductive thinking. Triangle ABC is equilateral; CD is an altitude. 1. What are m Aand m B? 2. What are m ACD and m BCD? C 3. What are m ADC and m BDC? 4. Is ACD BCD? Why? 5. Is AD BD? Why? A D B Note that altitude CD divides the equilateral triangle into two right triangles with acute angles of and. Look at just one of the triangles and compare the leg and the hypotenuse. What do you notice? Page 9 of 12

10 Let s see what else we can discover about triangles. Complete the chart with the lengths of the missing sides: Once again, keep your answer in the simplest radical form. Leg (a) Leg (b) Hypotenuse (c) x Work area Notice anything? Page 10 of 12

11 Theorem 7.9 Theorem 7.8 Chapter 7: Trigonometry... Sections Triangle Theorem In a triangle, the hypotenuse is times as long as either leg. The ratios of the side lengths can be written: Triangle Theorem In a triangle, the hypotenuse is as long as the leg (opposite the angle). The leg (opposite the angle) is times as long as the shorter leg. The ratios of the side lengths can be written: Page 461: 3 5, and more. Practice 7.4 Page 11 of 12

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