Grade 10 — Mathematics 10-3 › T3U05 The Pythagorean Theorem

Grade 10 › T3U05 The Pythagorean Theorem › L1
GRADE 10 · ALBERTAOUTCOME AB-MAT10-3-G2

L1 · Verifying the theorem; why right triangles only

Learning goal

“I can test whether the side lengths of a triangle fit a^2 + b^2 = c^2, I can show with a counterexample why the equality works only for right triangles, and I can explain why a 3:4:5 triangle must be a right triangle.”

Right triangle — a triangle with one angle of exactly 90 degrees. In trades language: two sides that are perpendicular, plumb or square with each other.
Hypotenuse — the side opposite the 90 degree angle. It is always the LONGEST side of a right triangle.
Leg — either of the two sides that form the 90 degree angle.
Pythagorean theorem — in a right triangle, the areas of the squares on the two legs add to the area of the square on the hypotenuse: a^2 + b^2 = c^2, where c is the hypotenuse.
Worked example

A triangle has legs 3 cm and 4 cm and hypotenuse 5 cm. The angle between the 3 cm and 4 cm sides is 90 degrees. Step 1 (build the squares): a square on the 3 cm leg has area 3^2 = 9 cm2. A square on the 4 cm leg has area 4^2 = 16 cm2. A square on the 5 cm hypotenuse has area 5^2 = 25 cm2. Step 2 (test the balance): 9 + 16 = 25. The two leg-squares exactly fill the hypotenuse-square. Say it plainly - the two linked representations: SQUARES model says "leg-square + leg-square = hypotenuse-square"; EQUATION model says the same thing in symbols: a^2 + b^2 = c^2. Every check in this lesson runs BOTH ways. Step 3 (the 3:4:5 fact): ANY triangle whose sides are in the ratio 3:4:5 (3, 4, 5 or 6, 8, 10 or 30, 40, 50...) is a right triangle, because the areas always balance: (3k)^2 + (4k)^2 = 25k^2 = (5k)^2. Rope-stretchers in ancient Egypt used a 12-knot rope in a 3-4-5 loop to lay out square field corners - the theorem at work thousands of years ago.

Check yourself — explain it

In the equation a^2 + b^2 = c^2, the letter c must always stand for...

Paired problem — example still visible

A right triangle has legs 6 cm and 8 cm and hypotenuse 10 cm. Check it with both models. -> SQUARES: 36 + 64 = 100 cm2. EQUATION: 6^2 + 8^2 = 36 + 64 = 100 = 10^2, so the areas balance.

Faded completion — last step is yours

"Legs 5 cm and 12 cm, hypotenuse 13 cm: 5^2 + 12^2 = 25 + ___ = ___, and 13^2 = ___, so the areas balance." -> 144; 169; 169

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