The Converse
Part of the Triangles section of Coddy's Geometry journey. Lesson 33 of 63.
The Pythagorean theorem says: if a triangle has a right angle, then a^2 + b^2 = c^2. Turned around, it is just as true: if the sides of a triangle satisfy a^2 + b^2 = c^2, then the triangle has a right angle, facing the longest side. This is the converse of the theorem.
It means you can test for a right angle with nothing but the three lengths. Builders use it to set out square corners: mark a rope into lengths of 6, 8 and 10, pull it tight into a triangle, and the corner between the 6 and the 8 is square, because 36 + 64 = 100.
Why it holds: a triangle is fixed by its three sides. A right triangle with legs 6 and 8 has hypotenuse 10, so a triangle with sides 6, 8 and 10 is that same right triangle.
Challenge
AB is 4 long. Drag C until BC = 3 and AC = 5, watching the readout. Look at the angle at B when you get there, then press Check.
Try it yourself
Drag the figure and watch
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 ReachingThe Range of the Third SideThe FormulaRecap: The Third Side5The Pythagorean Theorem
Squares on the SidesFinding the HypotenusePythagorean TriplesWhen It Is Not WholeRecap: The Hypotenuse3Perimeter
Around the EdgeA Missing SideIsosceles PerimeterSides as ExpressionsRecap: Around the Edge6Finding a Leg
Working BackwardsThe LadderWhich Side Is c?A Leg That Is Not WholeRecap: Finding a Leg