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# Area of Triangles

You know that Area = base x height / 2, but what other ways are there to find the area of a triangle? Brace yourself for some potent formulae.

# Area of Triangles - Shoelace Formula

Let $$A=(5,2)$$ and $$B=(6,-5)$$ be two vertices of triangle $$ABC.$$ If the area of the triangle is $$9,$$ which of the following are possible coordinates for the third vertex?

Consider a square $$ABCD$$ with side length $$19.$$ If we let $$M$$ and $$N$$ be the midpoints of $$\overline{BC}$$ and $$\overline{CD},$$ respectively. What is the area of $$\triangle AMN?$$

Statement $$1$$: If the area of the triangle bounded by the lines $$y=x, x+y=8$$ and the line through $$P=(h,k)$$ parallel to the $$x$$-axis is $$4h^2,$$ then $$P$$ lies on either of the two lines represented by $$4x^2+8y-y^2-16=0.$$

Statement $$2$$: The area of the triangle bounded by the lines $$y=x, x+y=8$$ and the $$x$$-axis is equal to half of the area of the triangle bounded by the line $$x+y=8$$ and the two coordinate axes.

Which of the following is true of the above two statements?

Triangle $$ABC$$ has vertices $$A=(5, 6)$$ and $$B=(6, 3).$$ If its area is $$22$$ and vertex $$C$$ lies on the line $$y=x+1$$ in the first quadrant, what are the coordinates of $$C?$$

Given three points $\begin{array} &A=(-1,0), &B=(0,12), &C=(k, k),\end{array}$ where $$k\ge 2,$$ what is the value of $$k$$ that minimizes the area of $$\triangle ABC?$$

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