L1 · The Pythagorean theorem

Learning goal

“I can state the Pythagorean theorem, identify the hypotenuse in any described right triangle, and find the hypotenuse when I know both legs - exactly when the answer is a perfect square root, and to the nearest tenth when it is not.”

Hypotenuse — the longest side of a right triangle; it is always the side directly opposite (across from) the right angle - the slanted side that never touches the right-angle corner.

Leg — either of the two shorter sides of a right triangle; the legs are the sides that meet at the right angle.

Pythagorean theorem — for any right triangle, a^2 + b^2 = c^2, where a and b are the legs and c is the hypotenuse. In area words: the square built on the hypotenuse has exactly the area of the two squares built on the legs, added together.

Square of a number — the number multiplied by itself: 7^2 = 7 x 7 = 49. Squaring a side length gives the area of the square built on that side.

Worked example

A right triangle has legs 3 cm and 4 cm. The right angle sits between the two legs; the side opposite it is the hypotenuse. Step 1: Build the squares on the legs: 3^2 = 9 and 4^2 = 16. Step 2: The square on the hypotenuse has area 9 + 16 = 25, so c^2 = 25. Step 3: c = sqrt(25) = 5 cm. Say it plainly: you SQUARE the legs, ADD, then take the SQUARE ROOT. The theorem hands you c^2 first - never skip the root.

Check yourself — explain it

In a^2 + b^2 = c^2, the letter c always stands for...

Paired problem — example still visible

Legs 6 cm and 8 cm -> c^2 = 36 + 64 = 100 -> c = sqrt(100) = 10 cm

Faded completion — last step is yours

"Legs 1.5 m and 2 m: c^2 = 2.25 + ___ = ___; c = ___ m" -> 4; 6.25; 2.5 m (sqrt(6.25) = 2.5, since 2.5 x 2.5 = 6.25)

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

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