Question

An acute triangle has two sides measuring 8 cm and 10 cm. What is the best representation of the possible range of values for the third side, s?

2 < s < 18
6 < s < 12.8
s < 2 or s > 18
s < 6 or s > 12.8

Answer 1

Triangle inequality gives 2 < s < 18. For acuteness the square of the largest side must be less than the sum of the squares of the other two.

- If s ≤ 10 (largest side 10): 10^2 < 8^2 + s^2 ⇒ 100 < 64 + s^2 ⇒ s^2 > 36 ⇒ s > 6.
- If s > 10 (largest side s): s^2 < 8^2 + 10^2 ⇒ s^2 < 164 ⇒ s < √164 ≈ 12.806.

Combining gives 6 < s < √164 ≈ 12.8. So the correct choice is: 6 < s < 12.8.