Question

In this section, you will derive the formula A=1/2absinC for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

Answer 1

To derive the formula for the area of a triangle, \( A = \frac{1}{2}ab \sin C \), using an auxiliary line, we will start with a triangle \( ABC \) with sides \( a \), \( b \), and angle \( C \) opposite side \( c \).

### Step 1: Draw the Triangle

1. Draw triangle \( ABC \) with vertices \( A \), \( B \), and \( C \).
2. Label the sides as follows:
- Side \( a \) is opposite vertex \( A \).
- Side \( b \) is opposite vertex \( B \).
- Side \( c \) is opposite vertex \( C \).

### Step 2: Draw the Auxiliary Line

1. From vertex \( C \), draw a perpendicular line to side \( AB \) at point \( D \).
2. This height (perpendicular line) will be denoted as \( h \), which represents the height of triangle \( ABC \) from vertex \( C \) to side \( AB \).

### Step 3: Relationship between Area and Triangle Dimensions

The area \( A \) of triangle \( ABC \) can be expressed as:

\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]

Here, we can use \( AB \) as the base, so:

\[
A = \frac{1}{2} \times c \times h
\]

### Step 4: Relate the Height to the Angles and Sides

Using the definition of the sine function in triangle \( ABC \):

- The height \( h \) can be expressed in terms of angle \( C \):
\[
h = b \sin A = a \sin B
\]

Now, we need to express \( h \) in terms of \( a \), \( b \), and \( \sin C \).

From the triangle, we can derive \( h \) using \( h = b \sin C \), since \( C \) is the angle formed by sides \( a \) and \( b \):

1. Substitute \( h \) back into the area formula earlier:
\[
A = \frac{1}{2} c (b \sin A)
\]

2. However, we can derive this directly using \( a \) and \( b \):
- We can also write \( h \) in terms of the sine of angle \( C \):
\[
h = a \sin B
\]

### Step 5: Express Area in Terms of \( a \), \( b \), and \( C \)

Using the law of sines, we know that

\[
\frac{c}{\sin C} = \frac{a}{\sin A} = \frac{b}{\sin B}
\]

This relationship allows us to write side \( c \) in relation to \( a \), \( b \), and \( C\):

\[
c = \frac{a \cdot b \cdot \sin C}{a \cdot b / c} \to c = \frac{ab \sin C}{\sqrt{(\sin A)(\sin B)}}
\]

Returning to the area formula, we plug in our expression for \( h \):

\[
A = \frac{1}{2}ab \sin C
\]

### Conclusion

We derived the formula for the area of triangle \( ABC \):

\[
A = \frac{1}{2}ab \sin C
\]

This shows that the area of a triangle is half the product of two of its sides multiplied by the sine of the included angle.