The Formula
Part of the Triangles section of Coddy's Geometry journey. Lesson 12 of 71.
The rule can be written as a formula. If a and b are the two known sides, the third side c satisfies:
|a - b| < c < a + b
The absolute value removes any minus sign from the difference, so it does not matter which of a and b is written first. With a = 6 and b = 10, |6 - 10| = 4 and 6 + 10 = 16. So the third side must be greater than 4 and less than 16.
To use the formula, work out the absolute value first. |6 - 10| becomes |-4|, which is 4. Then work out the sum on the other side, which is 16. The chain becomes 4 < c < 16. If you write the sides the other way round, |10 - 6| is also 4, so the formula gives the same range either way.
Challenge
Two sides of a triangle are 9 and 4. Use the formula to find the limits on the third side x.
Find the range of x.
Try it yourself
Solve for x
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Triangles
1Sides and Names
Three Sides, Three CornersSorting by SidesNaming by AnglesTicks into EquationsThe Longest SideRecap: Sorting Triangles2The Triangle Inequality
Can Three Sticks Meet?Exactly ReachingHow Short Can It Be?How Long Can It Be?The Range of the Third SideThe FormulaWhole-Number SidesRecap: The Third Side5The Hypotenuse
Squares on the SidesFinding the HypotenusePythagorean TriplesWhen It Is Not WholeSimplifying the RootRecap: The Hypotenuse8Special Right Triangles
The 45-45-90 TriangleLegs from the HypotenuseThe 30-60-90 TriangleAll Three SidesWhich Triangle Is It?Recap: Special Triangles11Mixing Area and Pythagoras
Height of an IsoscelesArea from the HeightThe Equilateral TriangleRight Triangle PerimeterRecap: Two Tools3Perimeter
Around the EdgeA Missing SideIsosceles PerimeterEquilateral PerimeterSides as ExpressionsWrite the PerimeterRecap: Around the Edge6Finding a Leg
Working BackwardsThe LadderWhich Side Is c?A Leg That Is Not WholeRecap: Finding a Leg