Geometry – Obtuse Angled Triangles

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.