Question

How do you use Heron's formula to find the area of a triangle with sides of lengths `19`, `14`, and `14`?

Answer 1

The area is `~~ 97.693` (rounded to thousandth's place)

Explanation:

Heron's formula is shown here:

We know that the side lengths are `19, 14, and 14`, so let's first find `s`.

`S = (19 + 14 + 14)/2`

Simplify:
`S = 47/2 = 23.5`

Now let's plug it into the formula for the area:
`A = sqrt(23.5(23.5-19)(23.5-14)(23.5-14))`

Now simplify:
`A = sqrt(23.5(4.5)(9.5)(9.5))`

`A = sqrt(9543.9375)`

`A ~~ 97.693` (rounded to thousandth's place)

Hope this helps!

Answer 2

`16A^2 = (19+14+14)(-19+14+14)(19-14+14)(19+14-14) = 47(9)(19)(19)`

`A = 57/4 sqrt(47)`

Explanation:

The other answer perfectly answers the question asked. I will rephrase the question and then answer it to show some alternatives to Heron.

How do we find the area of a triangle with sides `19, 14, 14` or any other three sides?

That's an isosceles triangle, though we're not to depend on that for this problem.

Let's just start with the answer, which is called Archimedes' Theorem. It's a modern form of Heron.

Given triangle sides `a,b,c` then area `A` satisfies

`16A^2 = 4a^2b^2 - (c^2-a^2-b^2)^2`

Before we apply it, let's talk about it. We see it only depends on the squared sides, making it extra useful given vertex coordinates. (Two D coordinates can often be handled more expeditiously with the Shoelace Theorem.)

We also see the formula looks asymmetrical, but it must be symmetrical, giving the same result if we interchange `a,b,c;` obviously the area shouldn't change. That means there's a more symmetric-looking form. Can you find it?

We plug in the numbers to get our answer.

`16A^2 = 4(19)^2(14)^2 - (14^2 - 19^2 - 14^2)^2 = 19^2(4(14)^2-19^2) = 19^2(28-19)(28+19)= 19^2 3^2 47`

`A = 57/4 sqrt(47)`

That was is just one variant of Archimedes' Theorem/Heron's formula. The factored form is cool:

`16A^2 = (a+b+c)(-a+b+c)(a-b+c)(a+b-c)`

This form is probably the way to go with this problem:

`16A^2 = (19+14+14)(-19+14+14)(19-14+14)(19+14-14) = 47(9)(19)(19)`

Piece of cake. Both these alternatives avoid the fractions until the end, which makes them more more pleasant. The squared form is preferred given vertex coordinates.