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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.

Sine Rule

In triangle $$ABC,$$ $$AB=3,$$ $$AC = 12,$$ and $$\sin \angle BAC = 0.5.$$ What is the area of triangle $$ABC?$$

In $$\triangle ABC,$$ $$AC = 3$$ and $$\sin \angle ACB = \frac{2}{3}.$$ If the area of $$\triangle ABC$$ is $$9,$$ what is $$BC?$$

In triangle $$ABC$$ above, $$\overline{AD}$$ bisects $$\overline{BC}.$$ Given that $$AB = 12,$$ $$AD =8,$$ and $$\sin \angle BAD = \frac{1}{4},$$ what is the area of $$\triangle ABC?$$

In triangle $$ABC,$$ $$\overline{AD}$$ bisects $$\angle BAC,$$ $$\overline{AE}$$ is perpendicular to $$\overline{BC},$$ and $$F$$ and $$G$$ are the midpoints of $$\overline{AB}$$ and $$\overline{AC},$$ respectively. Given that $$AB = 16,$$ $$AC=13,$$ $$AH=13,$$ and $$AI=8,$$ and $$\angle BAC = 90^{\circ},$$ which of the following triangles has an area equal to half the area of $$\triangle ABC?$$

In triangle $$ABC,$$ $$AB = 3$$ and $$BC=6.$$ If the area of the triangle is $$6,$$ what is $$\sin \angle ABC?$$

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