Geometry Triangle Inequality
Triangle Inequality Theorem The triangle inequality theorem states that any side of a triangle is always less than the sum of the other two sides. A AB = 8.72 cm
AC+BC = 15.81 cm
AC = 5.46 cm
AB+BC = 19.07 cm
BC = 10.35 cm
AB+AC = 14.18 cm
C
B
More importantly, the sum of the lengths of any two sides, in particular the smallest two sides, of a triangle is always greater than the length of the third side. A AB = 8.72 cm AC = 5.46 cm BC = 10.35 cm
C
AB+AC = 14.18 cm B
Imagine the bottom angles’ vertices are hinges. And the sides of length 10 and 13 can pivot on those hinges. Because the sum of 10 and 13 is greater than 15, the sides of 10 and 13 will come together at a point above the base and a triangle is form.
Imagine the bottom angles’ vertices are hinges in the above example. And the sides of length 5 and 7 can pivot on those hinges. Because the sum of 5 and 7 is less than 15, the sides of 5 and 7 will not intersect to form a triangle.
Imagine the bottom angles’ vertices are hinges in the above example as well. And the sides of length 5 and 10 can pivot on those hinges. Because the sum of 5 and 10 is equal to 15, the sides of 5 and 10 will intersect, but they will intersect only after they are collapse to the bottom segment. This is not a triangle.
Can the following lengths be those of the three sides of a triangle? 1. 9, 10, 15 2. 3, 4, 5 3. 3, 4, 7 4. 3, 4, 10 5. 7, 7, 10 6. 7, 7, 20 7. 2, 7, 7st 8. 10, 10, 10 possible
Find the length of the third side. Given two sides, the length of the third side can be a range of values. The angle closes‌
AC = 9.9 cm AB = 8.3 cm
A
The angle opens‌
B
BC = 2.1 cm C
BC = 18.1 cm
B A AC = 9.9 cm AB = 8.3 cm
C
Find the length of the third side.
If the length of two sides of a triangle are 8 inches and 19 inches, then the length of the third sides is a range of values. The third side must be greater than the difference 19 – 8 or 11 and it must be less than the sum 19 + 8 or 27. The third side must be between 11 and 27 inches long.
Inequalities in One Triangle 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 smaller side. 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. A 84.4ď‚° 5.4 cm 56.5ď‚° B
7.1 cm
39.1ď‚° 8.5 cm
C
List the angles in order from largest to smallest. A
11.1 cm
B
5.1 cm
C
8.5 cm
List the sides in order from largest to smallest. A 81.4ď‚°
39.7ď‚° B
58.8ď‚° C