Geometry

# Area of Triangles - 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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