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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.

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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