In ∆ABC AB=BC=4cm and angle B=90° find the area of the triangle using heron's formula
Answers
Given data : In ∆ ABC, AB = BC = 4 cm and angle B = 90°.
To find : Find the area of the triangle using heron's formula.
Solution : Here,
- Length of AB = 4 cm
- Length of BC = 4 cm
- Length of AC = ?
Firstly we have to find out, length of AC.
A/C to given we know that, given triangle, ∆ABC is right angle triangle.
Hence, side AC is hypotenuse
Now, by Pythagoras theorem
➜ (Hypo)² = (first side)² + (second side)²
➜ (AC)² = (AB)² + (BC)²
➜ (AC)² = (4)² + (4)²
➜ (AC)² = 16 + 16
➜ (AC)² = 32
➜ AC = √32
➜ AC = 4√2 cm
Let, side AB, BC and AC be a, b and c respectively.
Now, we have to find out semi-perimeter "s".
➜ s = (a + b + c)/2
➜ s = (4 + 4 + 4√2)/2
➜ s = (8 + 4√2)/2
➜ s = 4 + 2√2
Now,
➜ Area of ∆ ABC = √{s (s - a) (s - b) (s - c)}
➜ Area of ∆ ABC = √{(4 + 2√2) (4 + 2√2 - 4) ( 4 + 2√2 - 4) (4 + 2√2 - 4√2)}
➜ Area of ∆ ABC = √{(4 + 2√2) (2√2 ) (2√2) (4 - 2√2)}
➜ Area of ∆ ABC = √{(4 + 2√2) (8) (4 - 2√2)}
➜ Area of ∆ ABC = √{(4 + 2√2) (32 - 16√2)}
➜ Area of ∆ ABC = √{(4 + 2√2) (32 - 16√2)}
➜ Area of ∆ ABC = √64
➜ Area of ∆ ABC = 8 cm²
Answer : Hence, the area of the ∆ ABC is 8 cm².