Question

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

Answer 1

Answer: 10

Step-by-step explanation:

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

$m_{\overline{AC}}=\frac{3-1}{6-2}=\frac{1}{2}\\m_{\overline{BC}}=\frac{3-7}{6-4}=-2$

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 $A=\frac{1}{2}bh$.

Using the distance formula,

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

So, the area is $\frac{1}{2}(2\sqrt{5})(2\sqrt{5})=\boxed{10}$