L2 · Applying Pythagoras: missing sides and real problems

Learning goal

“I can find a missing LEG by rearranging a^2 + b^2 = c^2 (the hypotenuse square minus the known leg square), and I can prove whether a triangle is right-angled by testing the converse of the Pythagorean theorem.”

Converse of the Pythagorean theorem — the theorem run backwards: IF a triangle's side lengths satisfy a^2 + b^2 = c^2 (with c the longest side), THEN the triangle is right-angled, with the right angle opposite the longest side. It turns the theorem into a test.

Diagonal — a straight segment joining opposite corners of a rectangle (or other polygon). A rectangle's diagonal is the hypotenuse of the two right triangles it creates.

Worked example

A right triangle has hypotenuse 10 cm and one leg 6 cm. Find the other leg. Step 1: The hypotenuse square is the SUM of the two leg squares: 100 = 36 + b^2. Step 2: A part is found by subtracting: b^2 = 100 - 36 = 64. Step 3: b = sqrt(64) = 8 cm. Sanity check: a leg must be SHORTER than the hypotenuse - 8 < 10. If your "leg" comes out longer than the hypotenuse, you added when you should have subtracted.

Check yourself — explain it

To find a leg, you subtract the squares instead of adding them because...

Paired problem — example still visible

Hypotenuse 5 m, leg 4 m -> b^2 = 25 - 16 = 9 -> b = 3 m

Faded completion — last step is yours

"Hypotenuse 10 cm, leg 8 cm: b^2 = 100 - ___ = ___; b = ___ cm" -> 64; 36; 6 cm

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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