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.