Question

The number of integers x for which x, 10 and 24 are the sides of an acute angled triangle is

• A. 2
• B. 3
• C. 4
• D. 9

Answer 1

The correct option is A

2

For an acute angled triangle,
l² <m² +n²

where l is the largest side length among the side lengths of the triangle, l, m and n.
We have 2 cases for the integer x.

Case I.X>24>10
X² <24²+10²
X² <676
⇒ x ≤25
Butx+10>24
X > 14
∴ x can take integer values from 15 to 25. Only 25>24. Hence, there
Is only 1 value of x in this case.
Case. II.24>X>10
24²<x²+10²
476<X²
⇒X≥22
Butx<24
Hence x can take values 22, 23.
There are two possibilities in this case.
Note :Also when x=24, we can have an isosceles acute angled triangle.