Tuesday
Tuesday: Four Triangles Prove It
Rocket cuts four right triangles from paper, each with legs 3 and 4. "Why does the rule work? Show me."
"Arrange them inside this square," Raven says. She has drawn a square with side 7, which is 3 plus 4.
Rocket slides a triangle into each corner, the legs along the edges. A tilted square appears in the middle.
"What do you notice about the middle square?" Raven asks. "Its sides are the hypotenuses. Four of them."
Nova hovers above the paper, her light filling the tilted square. "Big square, take away four triangles," she says.
"Forty-nine take away twenty-four," Rocket says. "Twenty-five. And 25 is 9 plus 16!"
Nova hums. "Now do it with letters. Then it works for every right triangle, not just this one."
The rearrangement proof
Take any right triangle with legs a and b and hypotenuse c. Draw a big square with side a + b.
Fit four copies of the triangle into the corners, legs along the edges. The space left is a tilted square with side c.
The big square's area is (a + b)². The four triangles cover 4 × ½ab = 2ab. So the tilted square is (a + b)² - 2ab.
Multiply out: (a + b)² = a² + 2ab + b². Take away 2ab and c² = a² + b².
- Draw a square with side a + b.
- Place four copies of the right triangle in the corners, each with a leg along an edge.
- The leftover tilted square has side c, so its area is c².
- Big square minus four triangles: (a + b)² - 2ab = a² + b². That equals c².
| Legs a, b | Big square (a + b)² | Four triangles 2ab | Tilted square c² | c |
|---|---|---|---|---|
| 3, 4 | 49 | 24 | 25 | 5 |
| 5, 12 | 289 | 120 | 169 | 13 |
The numbers check the picture. The letters prove it for every right triangle at once.
Finding a leg
If you know the hypotenuse and one leg, turn the rule around: b² = c² - a².
Hypotenuse 13, leg 5: 13² - 5² = 144, and the square root of 144 is 12.
- Read the question.
- Tap your answer.
The tilted square has area 25. How long is its side, the hypotenuse c?- Read the question.
- Tap your answer.
- Read the question.
- Tap your answer.
A right triangle has longest side 13 cm and one leg 5 cm. How long is the other leg, in centimeters?
A right triangle has longest side 10 cm and one leg 6 cm. How long is the other leg, in centimeters?
A right triangle has longest side 17 cm and one leg 8 cm. How long is the other leg, in centimeters?
A right triangle has legs 8 and 15 cm. How long is the longest side, in centimeters?| Statement | True or false? |
|---|---|
| The four triangles in the proof are all the same size. | ? |
| The tilted square in the middle has side c. | ? |
| The proof only works for legs 3 and 4. | ? |
| To find a leg, subtract the squares instead of adding them. | ? |
Four triangles proved the rule. Tomorrow is Build Lab: squares on graph paper, counted by hand.