A triangle can be checked whether acute or not if the measures of two of its angles are known.If the lengths of three sides of a triangle are known, is there a way to determine whether the triangle is an acute angle?

An inequality in the line of Pythagorean Identity is used to determine this.

**Test for finding an acute triangle:**A given triangle is an acute triangle if the sum of the squares of the smaller side is greater than the square of the largest side.

We can write this as a mathematical statement. Suppose a, b and c are the lengths of a triangle and c the largest side. Then Δ ABC is an acute triangle if,

a

^{2} + b

^{2} > c

^{2}.

### Solved Example

**Question: **The sides of a triangle were given as 4 cm 5 cm and 7 cm. Bob felt that
it was an acute triangle. Check using the Pythagorean inequality whether
Bob was right.

** Solution: **

Sum of the squares of the smaller sides = 4^{2} + 5^{2} = 16 + 25 = 41

Square of the Largest side = 7^{2} = 49.

41 < 49 ⇒ a^{2} + b^{2} < c^{2}

Hence the triangle was not an acute triangle and Bob was wrong in his guess.