Answer:
A
Step-by-step explanation:
Remember that the formula for the area of a triangle is:
[tex]A=\frac{1}{2}bh[/tex]
In this case, our base will be the distance from the origin point (0, 0) to (x₁, y₁).
Our height will be the distance from the origin point (0, 0) to (x₂, y₂).
So, let's find each of the distances using the distance formula:
[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2[/tex]
BASE:
Let's let (x₁, y₁) be itself and let's let (0, 0) be (x₂, y₂). Substitute this into the distance formula. This yields:
[tex]d=\sqrt{{(0-x_1)^2+(0-y_1)^2[/tex]
Simplify:
[tex]d=\sqrt{(-x)^2+(-y)^2[/tex]
We can remove the negative:
[tex]d=\sqrt{((-1)x_1)^2+((-1)y_1)^2[/tex]
[tex]d=\sqrt{x_1\!^2+y_1\!^2}[/tex]
And this is the length of our base.
HEIGHT:
Let' let (0, 0) be (x₁, y₁) and (x₂, y₂) be itself. Substitute them into our distance formula:
[tex]d=\sqrt{(x_2-0)^2+(y_2-0)^2[/tex]
Simplify. So, our height is:
[tex]d=\sqrt{(x_2)^2+(y_2)^2[/tex]
Therefore, substitute the base and the height for b and h in our equation yields:
[tex]A=\frac{1}{2}(\sqrt{(x_1)^2+(y_1)^2)}(\sqrt{(x_2)^2+(y_2)^2})[/tex]
We can combine the square roots by multiplying. So:
[tex]A=\frac{1}{2}\sqrt{(x_1\!^2+y_1\!^2)(x_2 \!^2+y_2\!^2)[/tex]
The answer choice that represents this is A.
So, our answer is A.
And we're done!
