Topic 5-7 Inequalities in one triangle

How many different triangles can we make using these six pieces? 2 1.What are your guesses? 2.What guess is too low? Too high? 3.What strategies are you using as you think about how many triangles you can make? 4.What is the relationship between the side lengths?

Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. a + b > c a + c > b b + c > a Example:Determine if it is possible to draw a triangle with side measures 12, 11, and > 17  Yes > 12  Yes > 11  Yes Therefore a triangle can be drawn.

Example Determine if it is possible to draw a triangle with side measures 13, 6, and > 7  Yes > 6  Yes = 13  No Therefore a triangle cannot be drawn.

Finding the range of the third side: Since the third side cannot be larger than the other two added together, we find the maximum value by adding the two sides. Since the third side and the smallest side cannot be larger than the other side, we find the minimum value by subtracting the two sides. Example: Given a triangle with sides of length 3 and 8, find the range of possible values for the third side. The maximum value (if x is the largest side of the triangle)3 + 8 > x 11 > x The minimum value (if x is not that largest side of the ∆)8 – 3 > x 5> x Range of the third side is 5 < x < 11.

Example The lengths of two sides of a triangle are 6 and 8. Write the inequality that represents the possible lengths for the third side. The maximum value (if x is the largest side of the triangle)6 + 8 > x 14 > x The minimum value (if x is not that largest side of the ∆)8 – 6 > x 2 > x Range of the third side is 2 < x < 14

Triangle Inequality The smallest side is across from the smallest angle. The largest angle is across from the largest side. AB = 4.3 cm BC = 3.2 cm AC = 5.3 cm 54  37  89  B C A

Triangle Inequality – examples… For the triangle, list the angles in order from least to greatest measure. C A B 4 cm 6 cm 5 cm