Answer:
A
Step-by-step explanation:
Remember that the formula for the area of a triangle is:
![A=\frac{1}{2}bh](https://tex.z-dn.net/?f=A%3D%5Cfrac%7B1%7D%7B2%7Dbh)
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:
![d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%28x_2-x_1%29%5E2%2B%28y_2-y_1%29%5E2)
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:
![d=\sqrt{{(0-x_1)^2+(0-y_1)^2](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%7B%280-x_1%29%5E2%2B%280-y_1%29%5E2)
Simplify:
![d=\sqrt{(-x)^2+(-y)^2](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%28-x%29%5E2%2B%28-y%29%5E2)
We can remove the negative:
![d=\sqrt{((-1)x_1)^2+((-1)y_1)^2](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%28%28-1%29x_1%29%5E2%2B%28%28-1%29y_1%29%5E2)
![d=\sqrt{x_1\!^2+y_1\!^2}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7Bx_1%5C%21%5E2%2By_1%5C%21%5E2%7D)
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:
![d=\sqrt{(x_2-0)^2+(y_2-0)^2](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%28x_2-0%29%5E2%2B%28y_2-0%29%5E2)
Simplify. So, our height is:
![d=\sqrt{(x_2)^2+(y_2)^2](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%28x_2%29%5E2%2B%28y_2%29%5E2)
Therefore, substitute the base and the height for b and h in our equation yields:
![A=\frac{1}{2}(\sqrt{(x_1)^2+(y_1)^2)}(\sqrt{(x_2)^2+(y_2)^2})](https://tex.z-dn.net/?f=A%3D%5Cfrac%7B1%7D%7B2%7D%28%5Csqrt%7B%28x_1%29%5E2%2B%28y_1%29%5E2%29%7D%28%5Csqrt%7B%28x_2%29%5E2%2B%28y_2%29%5E2%7D%29)
We can combine the square roots by multiplying. So:
![A=\frac{1}{2}\sqrt{(x_1\!^2+y_1\!^2)(x_2 \!^2+y_2\!^2)](https://tex.z-dn.net/?f=A%3D%5Cfrac%7B1%7D%7B2%7D%5Csqrt%7B%28x_1%5C%21%5E2%2By_1%5C%21%5E2%29%28x_2%20%5C%21%5E2%2By_2%5C%21%5E2%29)
The answer choice that represents this is A.
So, our answer is A.
And we're done!