Question

In the triangle below,
1) Find the length of BC

2) Find the are area of Δ ABC

Answer 1

A1. Since we have two sides (AC and AB) and the included angle (60°), we are going to use the law of cosines to find the length of BC:

We can conclude that the length of the side BC is .

2. To find the area of triangle ABC we are going to use Heron's formula:

where

is the area of the triangle

is the semi-perimeter of the triangle

But first, we are going to find the semi-perimeter of our triangle using the formula:

We can infer for our triangle and from our previous calculation that , , and . Lets replace those values to find the semi-perimeter of our triangle:

Answer 2

1. Since we have two sides (AC and AB) and the included angle (60°), we are going to use the law of cosines to find the length of BC:

`BC= √(AB^2+AC^2-2(AB)(AC)Cos \alpha )`

`BC= √(6^2+4^2-2(6)(4)Cos(60))`

`BC= √(36+16-48Cos(60))`

`BC= √(52-48Cos(60))`

`BC=2 √(7)`

We can conclude that the length of the side BC is
`2 √(7)`.

2. To find the area of triangle ABC we are going to use Heron's formula:
`A= √(S(S-A)(S-B)(S-C))`
where

`A` is the area of the triangle

`S` is the semi-perimeter of the triangle

But first, we are going to find the semi-perimeter of our triangle using the formula:
`S= (A+B+C)/(2)`
We can infer for our triangle and from our previous calculation that
`A=2 √(7)`,
`B=4`, and
`C=6`. Lets replace those values to find the semi-perimeter of our triangle:

`S= (A+B+C)/(2)`

`S= (2 √(7+4+6) )/(2)`

`S=7.65`

Finally, we can use Heron's formula to find the area of our triangle:

`A= \sqrt{7.65(7.65-2 √(7))(7.65-4)(7.65-6)}`

`A=10.42`

We can conclude that the area of our triangle is 10.42 square units.