Published via Ecency

Hello math bugs(🐞) and hivers(🐝)
I hope you are strong and stout & doing well in life.

Well come to another interesting geometric proof. We all know about Heron's formula used to find area of a triangle when three sides of it are given. I'll show you today how to drive it.

To prove it we need to know formula of Pythagoras and to do a little construction. At first, let's draw a perpendicular from a vertex to its opposite side of the ∆"ABC*. Here I draw AD which os perpendicular to BC. Suppose, the three sides a, b , c are opposite to the respective angles A, angle B and angle C and also BD= x and CD= (a-x). Check details in the following figure:

Now we gonna use Pythagoras' formula in each of the smaller triangle namely ∆ABD and ∆ACD. In case ∆ABD , we can say c² = x² + h² or h² = c² - x². Check the follwoing figure also:

From ∆ACD , we also can get b² = (a-x)² + h² or h² = b² - a² -x² + 2ax. Check it below:

Check Heron's formula carefully. As there is no x or h in it. The the formula is given by a, b and c only. To get rid of h we need to compare equation1 and equation2 and we can have the value of x. Check the following figure:

See, our moto is to get rid other variables except a, b, and c. What can we do now! If we replace the value of x in equation1 or 2, we will have an equation devoid of unwanted variables. As equation1 is simplier than equation2, we gonna use the former. Check the figure below:

We know the area of a triangle is given by 1/2 ×base × height. Here we have consider a as base and height h. Once we find vaule of h from above equation, problem is solved because the vaule of h we gonna obtained will be in terms of three sides( a,b & c) of the ∆ABC. let's get it done.

Now area will be 1/2× base × height =1/2× a×√(S(S-a)(S-b)(S-c) unit². Also check the following figure: