Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Survey

Document related concepts

Golden ratio wikipedia, lookup

History of trigonometry wikipedia, lookup

Line (geometry) wikipedia, lookup

Geometrization conjecture wikipedia, lookup

Trigonometric functions wikipedia, lookup

Rational trigonometry wikipedia, lookup

Reuleaux triangle wikipedia, lookup

History of geometry wikipedia, lookup

Integer triangle wikipedia, lookup

Transcript

Indirect Proof and Inequalities 5-5 in One Triangle 5-5 inInequalities One Triangle Warm Up Lesson Presentation Lesson Quiz HoltMcDougal GeometryGeometry Holt Indirect Proof and Inequalities 5-5 in One Triangle Objectives Apply inequalities in one triangle. Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle The positions of the longest and shortest sides of a triangle are related to the positions of the largest and smallest angles. Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle Example 1A: Ordering Triangle Side Lengths and Angle Measures Write the angles in order from smallest to largest. The shortest side is smallest angle is F. The longest side is , so the , so the largest angle is G. The angles from smallest to largest are F, H and G. Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle Check It Out! Example 1B Write the angles in order from smallest to largest. The shortest side is smallest angle is B. The longest side is , so the , so the largest angle is C. The angles from smallest to largest are B, A, and C. Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle Example 2A: Ordering Triangle Side Lengths and Angle Measures Write the sides in order from longest to shortest. mR = 180° – (60° + 72°) = 48° The largest angle is Q, so the longest side is . The smallest angle is R, so the shortest side is . The sides from longest to shortest are PR, QR, and QP Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle Check It Out! Example 2B Write the sides in order from longest to smallest. mE = 180° – (90° + 22°) = 68° The largest angle is F, so the longest side is The smallest angle is D, so the shortest side is . . The sides from longest to shortest are DE, DF, and EF Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle A triangle is formed by three segments, but not every set of three segments can form a triangle. Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle A certain relationship must exist among the lengths of three segments in order for them to form a triangle. Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle Example 1: Applying the Triangle Inequality Theorem Tell whether a triangle can have sides with the given lengths. Explain. 7, 10, 19 No—by the Triangle Inequality Theorem, a triangle cannot have these side lengths. Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle Example 1: Applying the Triangle Inequality Theorem Tell whether a triangle can have sides with the given lengths. Explain. 2.3, 3.1, 4.6 Yes—the sum of each pair of lengths is greater than the third length. Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle Check It Out! Example 1 Tell whether a triangle can have sides with the given lengths. Explain. 8, 13, 21 No—by the Triangle Inequality Theorem, a triangle cannot have these side lengths. Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle Example 2-1: Applying the Triangle Inequality Theorem Tell whether a triangle can have sides with the given lengths. Explain. n + 6, n2 – 1, 3n, when n = 4. Step 1 Evaluate each expression when n = 4. n+6 n2 – 1 3n 4+6 (4)2 – 1 3(4) 10 15 12 Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle Example 2-1 Continued Step 2 Compare the lengths. Yes—the sum of each pair of lengths is greater than the third length. Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle Check It Out! Example 2-2 Tell whether a triangle can have sides with the given lengths. Explain. t – 2, 4t, t2 + 1, when t = 4 Step 1 Evaluate each expression when t = 4. t–2 4–2 2 Holt McDougal Geometry 4t 4(4) 16 t2 + 1 (4)2 + 1 17 Indirect Proof and Inequalities 5-5 in One Triangle Check It Out! Example 2-2 Continued Step 2 Compare the lengths. Yes—the sum of each pair of lengths is greater than the third length. Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle Example 3A: Finding Side Lengths The lengths of two sides of a triangle are 8 inches and 13 inches. Find the range of possible lengths for the third side. Let x represent the length of the third side. Then apply the Triangle Inequality Theorem. x + 8 > 13 x>5 x + 13 > 8 x > –5 8 + 13 > x 21 > x Combine the inequalities. So 5 < x < 21. The length of the third side is greater than 5 inches and less than 21 inches. Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle Check It Out! Example 3B The lengths of two sides of a triangle are 6 cm and 10 cm. Find the range of possible lengths for the third side. Rule for finding the range of the third side (short cut): Maximum value- add two sides Minimum value- subtract two sides Let x represent the 10+6 = 16 10-6 = 4 third side 4 < x <16 Min < x < Max Combine the inequalities. So 4 < x < 16. The length of the third side is greater than 4 cm and less than 16 cm. Holt McDougal Geometry