In a right triangle, the two legs and the hypotenuse are tied together by one equation: a² + b² = c². This lesson uses that equation to find a missing side of a right triangle — exactly, in simplest radical form, or rounded to the nearest tenth — and then turns the equation around: given three lengths, the converse tests whether they form a right triangle at all, and a comparison of the squares classifies the triangle as right, acute, or obtuse.
The Pythagorean Theorem relates the three sides of a right triangle, and its converse turns the relationship around: given three lengths, you can test whether they form a right triangle at all. This lesson finds missing sides of right triangles — exactly, in simplest radical form, or rounded — and then uses the converse, including a comparison of squares, to classify triangles as right, acute, or obtuse.
What you are looking at: A right triangle drawn on a grid, with a square built outward on each of its three sides. The right angle is at A. The horizontal leg is a, the vertical leg is b, and the side opposite the right angle — the hypotenuse — is c. Each square is labeled with the square of its side: a², b², and c² are the areas of the three squares.
What to do: Drag the orange handle on point B to change leg a, and the purple handle on point C to change leg b. Each leg snaps to a whole number from 1 to 5, and dragging is limited to the grid shown, so a handle can always be grabbed again.
What to notice: At every position, the two smaller square areas add to the largest square's area. The readout below works the theorem in separate steps for the current legs: the theorem stated, the legs substituted, and then the solving. When the three side lengths are all whole numbers, the readout says so — that is a Pythagorean triple.
What to do: Watch the Pythagorean Theorem classify a triangle, one step at a time. The first two examples draw the lengths as segments; the last two give numbers only. At the end of each example, the measurements form the triangle, each on its side.
Watch the steps plays the work line by line; Next Step shows one more line; the example buttons move between the four examples.
What to do: Decide whether the three lengths form a right, acute, or obtuse triangle, or do not form a triangle at all. The biggest value always goes in for c: compare a² + b² with c² — equal means right, greater means acute, and smaller means obtuse. Use Next Set to work through the eight sets in order.
What to do: Each problem gives you a triangle with work to do. The first four problems ask for a missing side of a right triangle; the last two give three side lengths and ask for a classification. Work in four steps: (1) choose the equation or comparison to use to solve the problem, (2) work out the square the problem turns on, (3) work out the second value the problem turns on — a square root (simplified, or rounded to the nearest tenth, when the problem asks for that) for a missing side, or the sum of the squares of the two shorter sides for a classification — and (4) report the answer the question actually asks for. The diagram comes first, then the problem statement, then the step question. The bold gold highlight in the diagram always marks the side the current step asks about, and once a missing side's measure is found it is labeled on the diagram in blue. Each choice you pick animates a segment in the diagram along the highlighted side to the length that choice describes, whether the choice is right or wrong, so you can see what the choice means; a missed choice is shown as a dashed amber ghost. A first miss gives a hint and another try, with the missed choice removed. A second miss on a step names the correct choice and explains it, and the activity then moves on. When all four steps are complete, the full worked solution appears. Press New Item for the next problem: the six problems take turns in order, and the problem number is shown with the step question.
Ready to practice? Work through these six questions, one at a time. If your first answer is wrong, you will get a hint and another try; the full explanation appears once you answer correctly, or after a second miss. Your score counts first-try answers.
Pythagorean Theorem and its Converse — Mr. J Geometry. All activities on this page run entirely in your browser.