Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Euler angles wikipedia , lookup
Golden ratio wikipedia , lookup
Multilateration wikipedia , lookup
Perceived visual angle wikipedia , lookup
Reuleaux triangle wikipedia , lookup
History of trigonometry wikipedia , lookup
Rational trigonometry wikipedia , lookup
Euclidean geometry wikipedia , lookup
Trigonometric functions wikipedia , lookup
Incircle and excircles of a triangle wikipedia , lookup
Inequalities in One Triangle Geometry Objectives: • Use triangle measurements to decide which side is longest or which angle is largest. • Use the Triangle Inequality Objective 1: Comparing Measurements of a Triangle • In diagrams here, you may discover a relationship between the positions of the longest and shortest sides of a triangle and the position of its angles. . largest angle longest side shortest side smallest angle If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side. B 3 5 A C mA > mC D If one ANGLE of a E 60° 40° triangle is larger than another ANGLE, then the SIDE opposite the larger angle is F longer than the EF > DF side opposite the smaller angle. You can write the measurements of a triangle in order from least to greatest. Ex. 1: Writing Measurements in Order from Least to Greatest Write the measurements of the triangles from least to greatest. a. m G < mH < m J JH < JG < GH J 100° 45° H 35° G Ex. 1: Writing Measurements in Order from Least to Greatest Write the measurements of the triangles from least to greatest. 8 Q 7 5 b. QP < PR < QR m R < mQ < m P R P Exterior Angle Inequality • The measure of an exterior angle of a triangle is greater than the measure of either of the two non adjacent interior angles. • m1 > mA and m1 > mB A 1 C B Ex. 2: Using Theorem 5.10 • DIRECTOR’S CHAIR. In the director’s chair shown, AB ≅ AC and BC > AB. What can you conclude about the angles in ∆ABC? A B C Ex. 2: Using Theorem 5.10 Solution • Because AB ≅ AC, ∆ABC is isosceles, so B ≅ C. Therefore, mB = mC. Because BC>AB, mA > mC by Theorem 5.10. By substitution, mA > mB. In addition, you can conclude that mA >60°, mB< 60°, and mC < 60°. A B C Ex. 3: Constructing a Triangle a. 2 cm, 2 cm, 5 cm b. 3 cm, 2 cm, 5 cm c. 4 cm, 2 cm, 5 cm Solution: Try drawing triangles with the given side lengths. Only group (c) is possible. The sum of the first and second lengths must be greater than the third length. Ex. 3: Constructing a Triangle a. 2 cm, 2 cm, 5 cm b. 3 cm, 2 cm, 5 cm c. 4 cm, 2 cm, 5 cm 2 2 5 C D D 3 4 2 A 5 2 B A 5 B Triangle Inequality Theorem • The sum of the lengths of any two sides of a Triangle is greater than the length of the third side. AB + BC > AC AC + BC > AB AB + AC > BC A C B Ex. 4: Finding Possible Side Lengths • A triangle has one side of 10 cm and another of 14 cm. Describe the possible lengths of the third side • SOLUTION: Let x represent the length of the third side. Using the Triangle Inequality, you can write and solve inequalities. x + 10 > 14 x>4 10 + 14 > x 24 > x ►So, the length of the third side must be greater than 4 cm and less than 24 cm. Ex. 5: • Solve the inequality: AB + AC > BC. A x+ 2 B x+ 3 3x - 2 C (x + 2) +(x + 3) > 3x – 2 2x + 5 > 3x – 2 5>x–2 7>x