If the equation of the hypotenuse of a right - angled isosceles triangle is `3x+4y=4` and its opposite vertex is (2, 2), then the equations of the perpendicular and the base are respectively

Options

A. `7x+y=16 & x-7y+12=0`
B. `7x-y=12&x+7y=16`
C. `5x+y=12 & x-5y+8=0`
D. `x+5y=12 & 5x-y=8`

Answer

To solve the problem, we need to find the equations of the perpendicular and the base of a right-angled isosceles triangle, given the equation of the hypotenuse and the coordinates of the opposite vertex.

### Step-by-Step Solution:

1. **Identify the given information:**
- The equation of the hypotenuse is \(3x + 4y = 4\).
- The coordinates of the opposite vertex are \((2, 2)\).

2. **Convert the hypotenuse equation to slope-intercept form:**
- We can rewrite the equation \(3x + 4y = 4\) in the form \(y = mx + c\).
- Rearranging gives us:
\[ 4y = -3x + 4 \implies y = -\frac{3}{4}x + 1 \]
- Here, the slope \(m\) of the hypotenuse is \(-\frac{3}{4}\).

3. **Determine the slope of the perpendicular line:**
- The slope of the line perpendicular to the hypotenuse can be found using the negative reciprocal of the slope of the hypotenuse.
- Therefore, the slope \(m_p\) of the perpendicular line is:
\[ m_p = \frac{4}{3} \]

4. **Use the point-slope form to find the equation of the perpendicular line:**
- The point-slope form of a line is given by:
\[ y - y_1 = m(x - x_1) \]
- Substituting \(m = \frac{4}{3}\) and the point \((2, 2)\):
\[ y - 2 = \frac{4}{3}(x - 2) \]
- Simplifying this equation:
\[ y - 2 = \frac{4}{3}x - \frac{8}{3} \]
\[ y = \frac{4}{3}x - \frac{8}{3} + 2 \]
\[ y = \frac{4}{3}x - \frac{8}{3} + \frac{6}{3} \]
\[ y = \frac{4}{3}x - \frac{2}{3} \]

5. **Find the slope of the base line:**
- Since the triangle is isosceles and right-angled, the base will have the same slope as the hypotenuse.
- Thus, the slope of the base line is also \(-\frac{3}{4}\).

6. **Use the point-slope form to find the equation of the base line:**
- Again using the point-slope form with slope \(-\frac{3}{4}\):
\[ y - 2 = -\frac{3}{4}(x - 2) \]
- Simplifying this:
\[ y - 2 = -\frac{3}{4}x + \frac{3}{2} \]
\[ y = -\frac{3}{4}x + \frac{3}{2} + 2 \]
\[ y = -\frac{3}{4}x + \frac{3}{2} + \frac{4}{2} \]
\[ y = -\frac{3}{4}x + \frac{7}{2} \]

7. **Final equations:**
- The equation of the perpendicular line is:
\[ y = \frac{4}{3}x - \frac{2}{3} \]
- The equation of the base line is:
\[ y = -\frac{3}{4}x + \frac{7}{2} \]

### Summary of Results:
- The equations of the perpendicular and the base are:
- Perpendicular: \(y = \frac{4}{3}x - \frac{2}{3}\)
- Base: \(y = -\frac{3}{4}x + \frac{7}{2}\)