Without using Heron’s formula, find the area of the triangle.

Source: NCTM Mathematics Teacher, January 2006

**SOLUTION**

**
**Applying the Law of Cosines

Area of triangle

**Answer**: square units

**Alternative solution 1**

**
**Most textbooks state the Law of Sines as

We have a more definitive Law of Sines

= diameter of circumcircle

Let’s apply it to and

= diameter

= diameter

diameter

radius

In right triangle

Area of triangle

We calculate similarly the areas of the triangle and

Area of triangle

Area of triangle

Area of triangle

**Alternative solution 2**

**
**In right triangle

In right triangle

Comparing the value of from Eq. and Eq.

Adding the two equations

—————-

Substituting the value of in Eq.

Area of triangle