Question

How do you use Heron's formula to find the area of a triangle with sides of lengths `7`, `4`, and `9`?

Answer 1

`Area=13.416` square units

Explanation:

Heron's formula for finding area of the triangle is given by
`Area=sqrt(s(s-a)(s-b)(s-c))`

Where `s` is the semi perimeter and is defined as
`s=(a+b+c)/2`

and `a, b, c` are the lengths of the three sides of the triangle.

Here let `a=7, b=4` and `c=9`

`implies s=(7+4+9)/2=20/2=10`

`implies s=10`

`implies s-a=10-7=3, s-b=10-4=6 and s-c=10-9=1`
`implies s-a=3, s-b=6 and s-c=1`

`implies Area=sqrt(10*3*6*1)=sqrt180=13.416` square units

`implies Area=13.416` square units

Answer 2

`13.416. units`

Explanation:

Use Heron's formula:

Heron's formula:

`color(blue)(Area=sqrt(s(s-a)(s-b)(s-c))`

Where,

`color(brown)(a-b-c=sides,s=(a+b+c)/2=semiperimeter` `color(brown)(of` `color (brown)(triangle`

So,

`color(red)(a=7`

`color(red)(b=4`

`color(red)(c=9`

`color(red)(s=(7+4+9)/2=20/2=10`

Substitute the values

`rarrArea=sqrt(10(10-7)(10-4)(10-9))`

`rarr=sqrt(10(3)(6)(1))`

`rarr=sqrt(10(18))`

`rarr=sqrt180`

We can further simplify that,

`color(green)(sqrt180=sqrt(36*5)=6sqrt5~~13.416.units`