IMC 2005 Q11
11. If two of the sides of a right-angled triangle are 5 cm and 6 cm long, how many possibilities are there for the length of the third side?
- A 0
- B 1
- C 2
- D 3
- E 4
Intermediate Mathematical Challenge 2005, Question 11. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 2
The side of length 5 cm cannot be the hypotenuse of the right-angled triangle as it is shorter than the side of length 6 cm. If the 6 cm side is the hypotenuse, then the third side of the triangle has length \(\sqrt{11}\) cm. If the 6 cm side is not the hypotenuse, then the hypotenuse has length \(\sqrt{61}\) cm. These are the only two possibilities.
Solution © UK Mathematics Trust, from the IMC 2005 solutions.