# Euclidean Geometry - Triangles Problem Solving

This wiki is about problem solving on triangles. You need to be familiar with some (if not all) theorems on triangles.

#### Contents

## Sample Problems

Triangles \(ABC\) and \(CDE\) are equilateral triangles of the same size. If \(AC=10\) and \(\angle BCD=80\), find the area of triangle \(ABF\).

Since \(\angle ACB=60^\circ\), \(\angle ACD=60^\circ+80^\circ=140^\circ.\)

Since \(\triangle ACD\) is isosceles, \(\angle CAD=\angle ADC=\dfrac{180^\circ-140^\circ}{2}=20^\circ.\)

It then follows that \(\angle BAF=\angle BAC-\angle DAC=60^\circ-20^\circ=40^\circ.\)

By the exterior angle theorem, \(\angle BFA=20^\circ+60^\circ=80^\circ.\)

By applying sine law on \(\triangle BAF\), we have

\[\dfrac{BF}{\sin 40^\circ}=\dfrac{10}{\sin 80^\circ}\implies BF=\dfrac{\sin 40^\circ}{\sin 80^\circ}(10).\]

Note that the area of a triangle is half the product of two adjacent sides multiplied by the sine of the included angle. We have

\[A=\dfrac{1}{2}(AB)(BF)(\sin 60^\circ)=\dfrac{1}{2}(10)\left(\dfrac{\sin 40^\circ}{\sin 80^\circ}\right)(10)(\sin 60^\circ) \approx 28.263.\ _\square \]

## Challenge Problems

## See Also

**Cite as:**Euclidean Geometry - Triangles Problem Solving.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/euclidean-geometry-triangles-problem-solving/