Pythagoras' theorem practice questions and worked solutions

In a² + b² = c², the letter c is always the hypotenuse: the side opposite the right angle, and always the longest of the three. Orientation makes no difference, and this is where diagrams catch students out, because textbook triangles sit neatly with the right angle at the bottom left while assessment triangles are rotated deliberately.

The reliable habit is to find the right angle, then trace directly across the triangle to the side facing it, and label that side before writing anything else down. Students who label first make far fewer errors than students who assume the longest-looking side in the drawing is the hypotenuse, since diagrams are frequently not to scale.

When both shorter sides are known you add: c² = a² + b². When the hypotenuse and one shorter side are known you subtract: b² = c² − a². Adding in the second case is the single most common error in this topic, and it produces an answer longer than the hypotenuse, which is geometrically impossible.

That impossibility is a free check. Before writing a final answer, ask whether the value you have found is shorter than the hypotenuse when it should be, or longer than both shorter sides when it is the hypotenuse. A student who applies this check catches almost every arithmetic slip without redoing the working.

Certain whole-number combinations appear constantly: 3-4-5, 5-12-13, 8-15-17 and 7-24-25, along with every multiple of them. A triangle with sides 9, 12 and 15 is simply 3-4-5 scaled by three, and spotting that lets you write the answer down rather than reaching for a calculator.

Recognising triples is genuinely useful under time pressure, but it carries a trap. Students who have memorised them sometimes assume a triangle with sides 5, 12 and 14 must be right-angled because two of the numbers look familiar. Always verify with the theorem rather than by pattern-matching two of the three sides.

The converse of the theorem is also true: if the squares on the two shorter sides sum to the square on the longest, the triangle is right-angled. This is a distinct skill from finding a missing side, and questions asking you to prove or disprove a right angle are testing the converse specifically.

Do not confuse this with the triangle inequality, which says only that the two shorter sides must sum to more than the longest for a triangle to exist at all. That condition is about whether the shape can be drawn; Pythagoras is about whether it contains a right angle. Multiple-choice questions frequently offer the triangle inequality as a distractor.

The distance formula in coordinate geometry is Pythagoras applied to the horizontal and vertical gaps between two points, and it is usually taught as a new formula rather than as the same idea in different clothing. Students who see the connection have one thing to remember instead of two.

The theorem also underpins the exact values in trigonometry, the magnitude of a vector in senior courses, and three-dimensional problems where you apply it twice — once in the base plane and again using that result as one side of a vertical triangle. Two-step three-dimensional questions are a standard senior assessment item.