The extremities of a diagonal of a rectangle which are parallel to the diagonal are

Options

A. `3x-4y=pm5`
B. `3x+4y=pm10`
C. `3x+4y=pm25`
D. `6x-8y=pm25`

Answer

To solve the problem regarding the extremities of a diagonal of a rectangle that are parallel to the diagonal, we can follow these steps:

### Step 1: Understand the Rectangle and Its Diagonal
Let’s denote the rectangle as ABCD, where:
- A(0, 0)
- B(a, 0)
- C(a, b)
- D(0, b)

The diagonal we are interested in is AC.

**Hint:** Identify the coordinates of the rectangle's vertices to visualize the diagonal.

### Step 2: Determine the Equation of the Diagonal
The diagonal AC connects points A(0, 0) and C(a, b). The slope of the line AC can be calculated as:
\[ \text{slope} = \frac{b - 0}{a - 0} = \frac{b}{a} \]

Using point-slope form, the equation of line AC is:
\[ y - 0 = \frac{b}{a}(x - 0) \]
Thus, the equation simplifies to:
\[ y = \frac{b}{a}x \]

**Hint:** Use the slope to derive the equation of the diagonal.

### Step 3: Identify the Parallel Line
A line parallel to AC will have the same slope, which is \( \frac{b}{a} \). If we want to find the equation of a line parallel to AC that passes through a point (x₁, y₁), it can be written as:
\[ y - y_1 = \frac{b}{a}(x - x_1) \]

**Hint:** Remember that parallel lines share the same slope.

### Step 4: Find the Extremities of the Diagonal
To find the extremities of the diagonal that are parallel to AC, we need to determine points that lie on this parallel line. For instance, if we take a point (0, 0) as one extremity, the other extremity can be found by moving along the direction of the slope \( \frac{b}{a} \).

Let’s say we choose a distance \( d \) along this slope:
- The coordinates of the second extremity can be given as:
\[ (x_1 + d, y_1 + \frac{b}{a}d) \]

**Hint:** Use the slope to determine the coordinates of the second extremity based on a chosen distance.

### Step 5: Write the Final Equation
The final equation of the line parallel to AC that passes through a point can be expressed as:
\[ y = \frac{b}{a}x + c \]
where \( c \) is a constant determined by the specific point through which the line passes.

**Hint:** Ensure to substitute the values correctly to find the specific line equation.

### Conclusion
The extremities of a diagonal of a rectangle that are parallel to the diagonal can be determined by finding the equation of the diagonal and then constructing parallel lines using the same slope. The specific coordinates can be calculated based on the rectangle's dimensions and the chosen distance along the slope.