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Mastering the Pythagorean Theorem and Hinge Theorem Concepts

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oleg
19 diapositives · 343 Views · Aug 01, 2014

Chapter 5 focuses on critical geometry concepts, primarily the Pythagorean Theorem and the Hinge Theorem. It explains how to apply these theorems to solve problems involving right triangles and comparative side lengths. The chapter includes examples to find values of unknowns in triangle configurations, calculates side lengths of special triangles like 45°-45°-90° and 30°-60°-90°, and discusses Pythagorean triples. This foundational knowledge is essential for deeper explorations in geometry.

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  1. Chapter 5 Review ISHS-Mrs. Bonn Geometry-Semester 1

  2. Warm-Up • Go to formulas page and write out Pythagorean Theorem. • Leave room for the Pythagorean triples.

  3. Example 1A: Using the Hinge Theorem and Its Converse Compare Compare the side lengths in ∆ABCand ∆ADC. AB = AD AC = AC BC > DC By the Converse of the Hinge Theorem,

  4. Example 1B: Using the Hinge Theorem and Its Converse Compare EF and FG. Compare the sides and angles in ∆EFH angles in ∆GFH. EH = GH FH = FH By the Hinge Theorem, EF < GF.

  5. The Pythagorean Theorem is probably the most famous mathematical relationship. As you learned in Lesson 1-6, it states that in a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse.

  6. Example 1A: Using the Pythagorean Theorem Find the value of x. Give your answer in simplest radical form. a2 + b2 = c2 Pythagorean Theorem 22 + 62 = x2 Substitute 2 for a, 6 for b, and x for c. 40 = x2 Simplify. Find the positive square root. Simplify the radical.

  7. Example 1B: Using the Pythagorean Theorem Find the value of x. Give your answer in simplest radical form. a2 + b2 = c2 Pythagorean Theorem (x – 2)2 + 42 = x2 Substitute x – 2 for a, 4 for b, and x for c. x2 – 4x+ 4 + 16 = x2 Multiply. –4x+ 20 = 0 Combine like terms. 20 = 4x Add 4x to both sides. 5 = x Divide both sides by 4.

  8. Check It Out! Example 1a Find the value of x. Give your answer in simplest radical form. a2 + b2 = c2 Pythagorean Theorem 42 + 82 = x2 Substitute 4 for a, 8 for b, and x for c. 80 = x2 Simplify. Find the positive square root. Simplify the radical.

  9. Check It Out! Example 2 What if...? According to the recommended safety ratio of 4:1, how high will a 30-foot ladder reach when placed against a wall? Round to the nearest inch. Let x be the distance in feet from the foot of the ladder to the base of the wall. Then 4x is the distance in feet from the top of the ladder to the base of the wall.

  10. A set of three nonzero whole numbers a, b, and c such that a2 + b2 = c2 is called a Pythagorean triple.

  11. Example 1A: Finding Side Lengths in a 45°- 45º- 90º Triangle Find the value of x. Give your answer in simplest radical form. By the Triangle Sum Theorem, the measure of the third angle in the triangle is 45°. So it is a 45°-45°-90° triangle with a leg length of 8.

  12. Example 1B: Finding Side Lengths in a 45º- 45º- 90º Triangle Find the value of x. Give your answer in simplest radical form. The triangle is an isosceles right triangle, which is a 45°-45°-90° triangle. The length of the hypotenuse is 5. Rationalize the denominator.

  13. A 30°-60°-90° triangle is another special right triangle. You can use an equilateral triangle to find a relationship between its side lengths.

  14. Example 3A: Finding Side Lengths in a 30º-60º-90º Triangle Find the values of x and y. Give your answers in simplest radical form. 22 = 2x Hypotenuse = 2(shorter leg) 11 = x Divide both sides by 2. Substitute 11 for x.

  15. Example 3B: Finding Side Lengths in a 30º-60º-90º Triangle Find the values of x and y. Give your answers in simplest radical form. Rationalize the denominator. y = 2x Hypotenuse = 2(shorter leg). Simplify.

  16. Example 4: Using the 30º-60º-90º Triangle Theorem An ornamental pin is in the shape of an equilateral triangle. The length of each side is 6 centimeters. Josh will attach the fastener to the back along AB. Will the fastener fit if it is 4 centimeters long? Step 1 The equilateral triangle is divided into two 30°-60°-90° triangles. The height of the triangle is the length of the longer leg.

  17. Example 4 Continued Step 2 Find the length x of the shorter leg. 6 = 2x Hypotenuse = 2(shorter leg) 3 = x Divide both sides by 2. Step 3 Find the length h of the longer leg. The pin is approximately 5.2 centimeters high. So the fastener will fit.