The base and the altitude of a triangle are `(3x - 4y)` and `(6x + 5y)` respectively. Find its area.

Answer

To find the area of the triangle with a base of \( (3x - 4y) \) and an altitude of \( (6x + 5y) \), we can use the formula for the area of a triangle:

\[ \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} \]

### Step 1: Identify the base and height
- Base = \( 3x - 4y \)
- Height = \( 6x + 5y \)

### Step 2: Substitute the values into the area formula
\[ \text{Area} = \frac{1}{2} \times (3x - 4y) \times (6x + 5y) \]

### Step 3: Expand the expression
To expand \( (3x - 4y)(6x + 5y) \), we can use the distributive property (also known as the FOIL method for binomials):

\[ (3x - 4y)(6x + 5y) = 3x \cdot 6x + 3x \cdot 5y - 4y \cdot 6x - 4y \cdot 5y \]

Calculating each term:
- \( 3x \cdot 6x = 18x^2 \)
- \( 3x \cdot 5y = 15xy \)
- \( -4y \cdot 6x = -24xy \)
- \( -4y \cdot 5y = -20y^2 \)

Now, combine these results:
\[ (3x - 4y)(6x + 5y) = 18x^2 + 15xy - 24xy - 20y^2 \]

### Step 4: Combine like terms
Combine the \( xy \) terms:
\[ 15xy - 24xy = -9xy \]
So the expression simplifies to:
\[ 18x^2 - 9xy - 20y^2 \]

### Step 5: Multiply by \( \frac{1}{2} \)
Now, substitute back into the area formula:
\[ \text{Area} = \frac{1}{2} \times (18x^2 - 9xy - 20y^2) \]

Distributing \( \frac{1}{2} \):
\[ \text{Area} = \frac{1}{2} \times 18x^2 - \frac{1}{2} \times 9xy - \frac{1}{2} \times 20y^2 \]
This simplifies to:
\[ \text{Area} = 9x^2 - \frac{9}{2}xy - 10y^2 \]

### Final Answer
Thus, the area of the triangle is:
\[ \text{Area} = 9x^2 - \frac{9}{2}xy - 10y^2 \quad \text{(square units)} \]

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