Corner to corner of a square
A square tile has sides of 3 cm. How long is its diagonal? A diagonal is the line from one corner to the opposite corner.
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Step 1 of 4
- A square of side 3 has area 9. Its diagonal cuts it into 2 triangles.
- The tile's corner is 90°. So the diagonal is the long side
cof each triangle. - A square built on that diagonal holds 4 of those triangles.
- So the area of that square is 18, and
c × c= 18.
The rule
The diagonal is between 4 cm and 5 cm long. The square on the diagonal has twice the first square's area. That holds for any side a.
Rule: Take a right triangle with two equal sides a. Its long side has c² = 2a².
Equal sides of 3 give c² = 2 × 3 × 3 = 18. 4 × 4 = 16 is below 18. 5 × 5 = 25 is above 18. So c is between 4 and 5.
Equal side a | c² = 2 × a × a | c is between |
|---|---|---|
| 6 | 72 | 8 and 9 |
| 13 | 338 | 18 and 19 |
| 17 | 578 | 24 and 25 |
| 22 | 968 | 31 and 32 |
Working backwards: if c² = 200, then a × a = 200 ÷ 2 = 100. So a = 10.
Watch out!
c² = 2a² is not c = 2a. For a = 6, c² is 72. But 2a is 12, and 12 × 12 is 144.