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 \(\triangle ABC?\)

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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?

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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.

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

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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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