In this lesson:
By the end of this lesson, you should be able to:
The two shorter sides are 3 and 4. How long is the side across from the right angle?
Areas: . The two small squares fill the big one exactly.
A right triangle with legs 5 and 12:
The longest side is 13. The pattern holds again.
A right triangle with legs 1 and 1:
The longest side is — not a whole number, but the rule still holds.
For legs and and hypotenuse : .
The hypotenuse is always the side opposite the right angle — and the longest side.
Find the right angle first. The side across from it is .
We checked three triangles. But there are infinitely many right triangles.
How do we know it works for every one? We need a proof.
Four copies of our triangle, one per corner, inside a square of side .
Each side of the inner shape is a hypotenuse of one triangle — so every side has length .
Four equal sides. But is it a square, or just a rhombus?
At each corner, the two acute angles () leave .
The inner shape has four equal sides and four right angles.
A shape with both is a square. Could you explain this to a classmate?
The big square has side , so its area is:
The big square also equals four triangles plus the inner square:
Subtract from both sides:
Without the right angle, the acute angles don't sum to — the inner shape isn't a square, and the proof collapses.
A right triangle has legs 6 and 8. Find the hypotenuse.
A right triangle has hypotenuse 13 and one leg 5. Find the other leg.
Set up and solve. Try it before advancing.
From the diagram alone, explain in your own words why .
Cover all four moves: setup, inner square, two area calculations, simplify.
✓ Checking 3-4-5 shows the rule works for that triangle ✓ Rearranging area shows it must work for every right triangle Watch out: needs a right angle — and is always opposite it