The ends of the base of an isosceles triangle are at `(2a, 0)and (0, a).` The equation of one side is `x = 2a. `The equation of the other side, is

Options

A. x + 2y - a = 0
B. x + 2y = 2a
C. 3x + 4y - 4a = 0
D. 3x - 4y + 4a = 0

Answer

To find the equation of the other side of the isosceles triangle with given vertices, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the vertices of the triangle**:
The ends of the base of the isosceles triangle are given as \( A(2a, 0) \) and \( B(0, a) \). The third vertex, \( C \), is at \( (2a, H) \) where \( H \) is to be determined.

2. **Use the property of isosceles triangles**:
In an isosceles triangle, the lengths of the two sides from the base to the apex are equal. Therefore, we can set the lengths \( AC \) and \( BC \) equal:
\[ AC = BC \]

3. **Calculate the lengths**:
- The length \( AC \) can be calculated as:
\[ AC = \sqrt{(2a - 2a)^2 + (H - 0)^2} = \sqrt{H^2} = |H| \]
- The length \( BC \) can be calculated as:
\[ BC = \sqrt{(2a - 0)^2 + (H - a)^2} = \sqrt{(2a)^2 + (H - a)^2} \]

4. **Set the lengths equal**:
Since \( AC = BC \), we have:
\[ |H| = \sqrt{(2a)^2 + (H - a)^2} \]

5. **Square both sides to eliminate the square root**:
\[ H^2 = (2a)^2 + (H - a)^2 \]
Expanding the right side:
\[ H^2 = 4a^2 + (H^2 - 2aH + a^2) \]
Simplifying gives:
\[ H^2 = 4a^2 + H^2 - 2aH + a^2 \]
Cancel \( H^2 \) from both sides:
\[ 0 = 5a^2 - 2aH \]

6. **Solve for \( H \)**:
Factoring out \( a \):
\[ 0 = a(5a - 2H) \]
Since \( a \neq 0 \), we can set:
\[ 5a - 2H = 0 \implies 2H = 5a \implies H = \frac{5a}{2} \]

7. **Determine the coordinates of the third vertex**:
The coordinates of the third vertex \( C \) are \( (2a, \frac{5a}{2}) \).

8. **Find the slope of the line \( BC \)**:
The slope \( m \) of line \( BC \) from \( B(0, a) \) to \( C(2a, \frac{5a}{2}) \) is:
\[ m = \frac{\frac{5a}{2} - a}{2a - 0} = \frac{\frac{5a}{2} - \frac{2a}{2}}{2a} = \frac{\frac{3a}{2}}{2a} = \frac{3}{4} \]

9. **Use point-slope form to write the equation of line \( BC \)**:
Using point \( B(0, a) \):
\[ y - a = \frac{3}{4}(x - 0) \]
Rearranging gives:
\[ y - a = \frac{3}{4}x \implies y = \frac{3}{4}x + a \]

10. **Convert to standard form**:
Rearranging the equation:
\[ 3x - 4y + 4a = 0 \]

### Final Answer:
The equation of the other side of the triangle is:
\[ 3x - 4y + 4a = 0 \]