Use Inequalities in A Triangle

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Presentation transcript:

Use Inequalities in A Triangle

Objectives: Use triangle measurements to decide which side is longest or which angle is largest. Use the Triangle Inequality Theorem

Comparing Measurements of a  shortest side smallest angle longest side largest angle The longest side and largest angle of a  are opposite each other. The shortest side and smallest angle of a  are opposite each other.

Theorem 5.10 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. mA > mC

Theorem 5.11 If one ANGLE of a triangle is larger than another ANGLE, then the SIDE opposite the larger angle is longer than the SIDE opposite the smaller angle. 60° 40° EF > DF

Example 1: Writing Measurements in Order from Least to Greatest Write the measurements of the triangles from least to greatest. m G < mH < m J JH < JG < GH 100° 45° 35°

Example 2: Writing Measurements in Order from Least to Greatest Write the measurements of the triangles from least to greatest. QP < PR < QR m R < mQ < m P 8 5 7

Using the Triangle Inequality Not every group of three segments can be used to form a triangle. The lengths of the segments must fit a certain relationship.

Activity: Constructing a Triangle 2 cm, 2 cm, 5 cm 3 cm, 2 cm, 5 cm 4 cm, 2 cm, 5 cm Activity: Let’s try drawing triangles with the given side lengths. Blue straw: 2cm Green straw: 3 cm Red straw: 4cm Yellow straw: 5cm

Activity: Constructing a Triangle 2 cm, 2 cm, 5 cm 3 cm, 2 cm, 5 cm 4 cm, 2 cm, 5 cm Notice, only group (c) is possible. Thus, what we can deduce is that the sum of the first and second lengths must be greater than the third length.

Theorem 5.12: 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

Example 3: 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. If x was the smallest side, then x + 10 > 14, so x > 4 If x was the longest side, then 10 + 14 > x, so 24 > x ► So, the length of the third side must be greater than 4 cm and less than 24 cm.

Example 4: Using Algebra to Find Possible Side Lengths Solve the inequality: AB + AC > BC. (x + 2) +(x + 3) > 3x – 2 2x + 5 > 3x – 2 5 > x – 2 7 > x

Assignment Handout on Triangle Inequalities