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Understanding Triangle Inequalities: Rules and Examples

Learn how to determine if a set of values can form a triangle, find the third side's range, and order segments or angles correctly. Explore examples and the Triangle Inequality Theorem. Test your knowledge with practical exercises.

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Understanding Triangle Inequalities: Rules and Examples

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  1. Triangle Inequalities Lesson 7.3 & 7.4

  2. Objectives • Determine if a set of values can form a triangle. • Determine the range of value for the 3rd side. • Determine the order of line segments or angle measurements.

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

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

  5. B 8x-10 7x+6 7x+8 A C Triangle Inequality – examples… For the triangle, list the sides in order from shortest to longest measure. (7x + 8) ° + (7x + 6 ) ° + (8x – 10 ) ° = 180° 22 x + 4 = 180 ° 22x = 176 X = 8 m<C = 7x + 8 = 64 ° m<A = 7x + 6 = 62 ° m<B = 8x – 10 = 54 ° 54 ° 62 ° 64 °

  6. Converse Theorem & Corollaries Converse: If one angle of a triangle is larger than a second angle, then the side opposite the first angle is larger than the side opposite the second angle. Corollary 1: The perpendicular segment from a point to a line is the shortest segment from the point to the line. Corollary 2: The perpendicular segment from a point to a plane is the shortest segment from the point to the plane.

  7. 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. 12 + 11 > 17  Yes 11 + 17 > 12  Yes 12 + 17 > 11  Yes Therefore a triangle can be drawn.

  8. Can these lengths be the sides of a triangle? • 4, 7, 10 • 3, 5, 8 • 3, 5, 7.9 • Yes • No • Yes There is a shortcut!!! (Use s + m > l is enough.) Ex: s + m > l 4 + 7 > 10 (True, they form a triangle! Done!) Ex: s + m > l 3 + 5 = 8 (No, they collapse into a straight line.)

  9. 6 ? 9 9 6 ? Two sides of a triangle measure 6 cm and 9 cm. Write an inequality that represents the range of values for the possible lengths of the third side. x + 6 > 9 x > 3 6 + 9 > x 15 > x Range of the third side is 3 < x < 15.

  10. 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.

  11. What’s the third side? • If the lengths of two sides of a triangle measure 7 and 12, the length of the third side could measure Case 1: finding the longest side. Case 2: 12 being the longest side. m + s > l 7 + x > 12 x > 5 m + s > l 7 + 12 > x 19 > x Range of the third side is 5 < x < 19.

  12. You do it!! • 9 and 15 6 < x <24 • 11 and 20 9 < x < 31 • 23 and 14 9 < x < 37

  13. Exit Ticket: • Decide whether each set of numbers is a triangle. 1) 15,12,9 2) 20, 10, 9 • Find the range of the third side if the other two are 5 and 8.

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