Pythagorean Theorem - Interactive Activity

Definitions and Main Ideas

The Pythagorean Theorem applies to right triangles only: the squares of the two legs add to the square of the hypotenuse. It finds a missing side when the other two are known — a leg is found by subtracting, the hypotenuse by adding. Whole-number side sets that satisfy the theorem are Pythagorean triples, and every triple belongs to a family generated by a root triple.

Vocabulary

Right Triangle

A triangle that contains one right angle (an angle measuring 90°).

Legs

The two sides of a right triangle that form the right angle.

Hypotenuse

The side of a right triangle opposite the right angle. The hypotenuse is always the longest side of the triangle.

Pythagorean Theorem

In any right triangle, the sum of the squares of the measures of the legs is equal to the square of the measure of the hypotenuse: a² + b² = c², where a and b are legs and c is the hypotenuse.

Pythagorean Triple

Three whole numbers that satisfy the Pythagorean Theorem, such as 3, 4, 5 or 5, 12, 13.

Root Triple

A Pythagorean triple whose numbers have a greatest common factor of 1. Every multiple of a root triple is another triple in the same family.

Simplest Radical Form

A radical is in simplest radical form when the number under the radical sign has no perfect-square factor other than 1. For example, √52 = √(4 · 13) = 2√13, so 2√13 is the simplest radical form of √52. A simplified radical keeps the exact value.

Main Ideas

1. The theorem is only for right triangles, and c always names the hypotenuse — identify it before substituting.
2. Finding the hypotenuse: add the squares of the legs, then take the square root. Finding a leg: subtract the square of the known leg from the square of the hypotenuse, then take the square root.
3. Answers may be left in simplest radical form; a simplified radical keeps the exact value. When a problem asks for a decimal, round to the nearest tenth.
4. Recognizing common triples (3-4-5, 5-12-13, 7-24-25, 8-15-17) and their multiples speeds up every problem.
5. Always check that the hypotenuse comes out longest, and that lengths are positive.

Squares on the Sides Explorer

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: a² + b² = c². When the three side lengths are all whole numbers, the readout says so — that is a Pythagorean triple.

Build the Missing Side

What to do: Each problem gives you a right triangle with one side missing. Work in four steps: (1) choose the equation to use to solve the problem, (2) work out the square of the missing side, (3) take the square root — simplifying the radical, or rounding to the nearest tenth, when the problem asks for that — 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 the 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.

Triple Families

What you are looking at: A right triangle whose sides are a Pythagorean triple, with the check worked in the readout: the theorem stated, the triple substituted, and the two sides of the equation compared. Every triple belongs to a family — the multiples of one root triple.

Is It a Triple?

What to do: Each candidate below gives three numbers. Decide whether they form a Pythagorean triple. A first miss gives a hint and another try; a second miss names the answer and shows the full check, worked the same way as the readout above: theorem stated, numbers substituted, sides compared. Press Next Candidate to move on — the six candidates take turns in order.