Sorting by the Test
Part of the Triangles section of Coddy's Geometry journey. Lesson 43 of 71.
Here are the steps of the test from start to finish. Sort the sides so the longest is last. Square all three. Add the first two squares, then subtract the third. Read the sign of the result. For sides 7, 9 and 12:
7^2 + 9^2 - 12^2
49 + 81 - 144
-14
The result is negative, so the triangle is obtuse. Its largest angle faces the side of length 12.
One test answers three questions: is the triangle right, is it acute, or is it obtuse. You never need to measure an angle. Together with the triangle inequality from an earlier lesson, three lengths tell you everything. First check whether they form a triangle at all. Then run the test to find out what kind of triangle it is.
Challenge
A triangle has sides 4, 5 and 6. Run the test: add the squares of the two shorter sides, then subtract the square of the longest side.
Work the expression out fully. The sign of the result tells you the kind of triangle.
Try it yourself
Simplify the expression
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