Question

What is the area of triangle ABC?
A(2,1) B(4,7) C(6,3)

Answer 1

Answer: 10

Explanation:

It appears that
`\overline{AC} \perp \overline{BC}`, so to check if this is the case, we can use the slope formula.

`m_{\overline{AC}}=(3-1)/(6-2)=(1)/(2)\\m_{\overline{BC}}=(3-7)/(6-4)=-2`

Since these slopes are negative reciprocals, we can see that this is the case. So, we can take
`\overline{AC}` as the base of the triangle and
`\overline{BC}` as the height of the triangle, and then use the formula
`A=(1)/(2)bh`.

Using the distance formula,

`AC=\sqrt{(6-2)^(2)+(3-1)^(2)}=√(16+4)=2√(5)\\BC=\sqrt{(4-6)^(2)+(7-3)^(2)}=√(4+16)=2√(5)`

So, the area is
`(1)/(2)(2√(5))(2√(5))=\boxed{10}`