How many obtuse-angled triangles can you form with 10,24,x as the sides where x is an integer?
Answer: 14
Explanation:
For three numbers to form an obtuse-angled triangle the square of the largest number should be greater than the sum of the squares of the other two numbers.
For three numbers to form a triangle the sum of any two numbers should be greater than the third.
Hence, if x is the largest number, then
x2 > 102+ 242
x2 > 676
Hence x > 26 – – – – – – – – – – (1)
If 24 is the largest number, then
242 > x2+ 102
x2 < 476
Hence x < 22 – – – – – – – – – – (2)
Also, for 10,24,x to form a triangle,
10 + 24 > x
x < 34 – – – – – – – – – – (3)
Also,
10 + x > 24
x > 14 – – – – – – – – – – (4)
Now combining (1) and (3), we see that
x can be 27,28,29,30,31,32,33 – – – 7 values
Combining (2) and (4), we see that
x can be 15,16,17,18,19,20,21 – – – 7 values
For each value of x we get a distinct triangle. Hence 14 triangles are possible.