Question

An acute triangle has sides measuring 10 cm and 16 cm. The length of the third side is unknown.

Which best describes the range of possible values for the third side of the triangle?

x < 12.5, x > 18.9
12.5 < x < 18.9
x < 6, x > 26
6 < x < 26

Answer 1

Triangle inequality gives 6 < x < 26. For an acute triangle the square of the largest side must be less than the sum of the squares of the other two sides.

- If x ≥ 16 (so x is largest): x^2 < 10^2 + 16^2 = 356 ⇒ x < √356 ≈ 18.87.
- If x ≤ 16 (so 16 is largest): 16^2 < 10^2 + x^2 ⇒ x^2 > 156 ⇒ x > √156 ≈ 12.49.

Combining with 6 < x < 26 gives √156 < x < √356, i.e. approximately 12.5 < x < 18.9.

So the correct choice is: 12.5 < x < 18.9.