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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.

Problem Solving

If the lengths of the three sides of triangle \(\triangle ABC\) sum to \(13,\) and the inscribed circle has radius \(8,\) what is the area of triangle \(\triangle ABC?\)

Consider a triangle inscribed in a circle with radius \(2.\) If one side of the triangle is a diameter of the circle, what is the largest possible area of the triangle?

In the above diagram, we are given two side lengths \[\rvert \overline{AB} \lvert = 3, \rvert \overline{AC} \lvert = 9.\] If \(\sin(\angle B+ \angle C) = \frac{1}{4},\) what is the area of \(\triangle ABC?\)

Note: The above diagram is not drawn to scale.

In the above right triangle, if \[\angle ABC = 30^{\circ}, \angle ADC = 45^{\circ}, \lvert \overline{BC} \rvert = 14,\] what is the area of \(\triangle ADC?\)

Triangle \(ABC\) has vertices \(A=(-4,k),\) \(B=(5,0)\) and \(C=(4,6).\) If the area of triangle \(ABC\) is \(23\), what are the possible values of \(k\)?

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