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# FM Herons Formula

## 1.6 Solving Triangles in 2 and 3 dimensions

Note: if you cannot remember the trigonometric identities, revise notes for 1.5 Methods for Solving Triangles.

### Solving for Area without Height

• In some cases, triangles will be given with the edge lengths rather than the height. In this case our regular formula for the area of a triangle (A=\frac{1}{2} b h) needs some reworking. If we draw a line down the middle of a triangle, we can effectively split it into two right-angled triangles from which we can use our trigonometric identities to solve for the height. Using our formula for sine, we find:

\sin (A) =\frac{h}{c}

h =c \sin (\theta)

Consequently:

A=\frac{1}{2} b c \sin (\theta)

• This formula works for non-right angled triangles as well.

Note: trigonometry is explored in more detail in notes 2.3 Solving Triangles using Trigonometry and 2.4 Applications of Trigonometry and Pythagoras Theorem.

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## 1.5 Methods for Solving Triangles

### The 180o Rule

• For any given triangle, the sum of the inside angles is equal to 180o:

A+B+C=180^{\circ}

Where A, B and C are the inside angles of a triangle.

Example

We wish to find the unknown angle, X, in the above triangle. As the other angles are known, we can use the 180o rule. First, we substitute in the known values:

X+35+45=180

Now we solve for the unknown angle:

X+80 &=180

X &=100^{\circ}

### The Sine Rule

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