proof of heron formula
Answers
Answer:
Step-by-step explanation:
Hero’s Formula For Triangle
According to Heron, we can find the area of any given triangle, whether it is a scalene, isosceles or equilateral, by using the formula, provided the sides of the triangle. Suppose, a triangle ABC, whose sides are a, b and c, respectively. Thus, the area of a triangle can be given by;
Heron's formula
Where “s” is semi-perimeter = (a+b+c) / 2
And a, b, c are the three sides of the triangle.
Example: A triangle PQR has sides a=4, b=13 and c=15. Find the area of the triangle.
Semiperimeter of triangle PQR, s = (4+13+15)/2 = 32/2 = 16
By heron’s formula, we know;
A = √[s(s-a)(s-b)(s-c)]
Hence, A = √[16(16-4)(16-13)(16-15)] = √(16 x 12 x 3 x 1) = √576 = 24
This formula is applicable to all types of triangles. Now let us derive the area formula given by Heron.
Heron’s Formula For Quadrilateral
Let us learn here to find the area of quadrilateral using Heron’s formula.
If ABCD is a quadrilateral, where AB//CD and AC & BD are the diagonals.
AC divides the quad.ABCD into two triangles ADC and ABC.
And h1 and h2 are the perpendicular drawn from D and B to the diagonal AC at point O1 and O2 respectively.
Area of Quad. ABCD = Area of ∆ADC + Area of ∆ABC
= 1/2 x AC x h1 + 1/2 x AC x h2
= 1/2 x AC (h1 + h2)
Area of quadrilateral = 1/2 x One diagonal x Sum of the altitude of the triangles formed when diagonal AC divided the quadrilateral ABCD.
This was the general formula, but if we want we can use Heron’s formula here to find the areas of the two triangles, by knowing the length of their sides.
Heron’s Formula for Equilateral Triangle
As we know the equilateral triangle have all its sides equal. To find the area of equilateral triangle let us first find the semi perimeter of equilateral triangle will be:
s = (a+a+a)/2
s=3a/2
where a is the length of the side.
Now, as per the heron’s formula, we know;
Heron's formula for equilateral triangle
Since, a = b = c
Therefore,
A = √[s(s-a)3]
which is the required formula.
Heron’s Formula Proof
There are two methods by which we can derive the Hero’s formula. First by using trigonometric identities and cosine rule. Secondly by the solving a