Direct road, roundabout road
Home to school is 10 km by the direct road. Going by the shop is 6 km, then 8 km. That is the roundabout way. Which way is shorter?
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- Home and school: 10 km by the direct road.
- By the shop: 6 km, then 8 km.
- Roundabout: 6 + 8 = 14 km.
- Direct 10 is less than 14. Direct is shorter.
The rule
The direct road is shorter: 10 km is less than 14 km.
A road with a bend is longer than the direct road. So each side is shorter than the other two added. Checking the longest side is enough. A third side is longer than the difference of the other two. It is shorter than their sum.
If the shop is on the direct road, both lengths are equal. No triangle is shut in.
Take sides 7 and 12. The difference is 12 − 7 = 5. The sum is 7 + 12 = 19.
| Two sides | The third side is more than | And less than |
|---|---|---|
| 7 and 12 | 5 | 19 |
| 4 and 4 | 0 | 8 |
Why is the longest side enough? Any other side is no longer than the longest side. So it is shorter than the longest side plus the remaining side.
Watch out!
"The third side can be anything shorter than the other two added." Not so. With 7 and 12, a 4 cm stick is too short. 4 + 7 = 11, which is less than 12.