Verify Heron's formula for area of a triangle.
Answers
Area of a triangle: 1/2*base*height.
Area of a triangle (Heron's Formula): √s(s - a)(s - b)(s - c).
Let's suppose the base of a triangle is 2cm and height is 4cm.
Area of a triangle = 1/2*2*4 = 4cm^2.
The third side: Hypotenuse.
Hypotenuse^2 = Base^2 + Height^2 = 2^2 + 4^2 = 4 + 16 = 20cm
Hypotenuse = √Base + Height
Hypotenuse = √20 = 2√5cm
Heron's formula.
Let a be 2cm, b be 4cm and c be 2√5cm.
a + b + c = Perimeter.
2 + 4 + 2√5 = Perimeter.
6 + 2√5cm = Perimeter.
Semi - Perimeter = 6 + 2√5/2 = 2(3 + √5)/2 = 3 + √5cm.
s - a = 3 + √5 - 2 = 1 + √5
s - b = 3 + √5 - 4 = -1 + √5
s - c = 3 + √5 - 2√5 = 3 - √5
Area = √s(s - a)(s - b)(s - c)
= √(3 + √5)*(1 + √5)* (-1 + √5)*(3 - √5)
=√(3^2 - √5^2)*(-1^2 + √5^2)
= √(9 - 5)(-1 + 5)
= √4*4
= √16
= 4cm^2
Area of triangle = Area of triangle (Heron's Formula).
Hence verified.
Hope it helps :)