A straight line passes through the points (5 , 0) and (0,3) . The length of the perpendicular from the point (4 , 4) on the line is

Options

A. `(sqrt(17))/(2)`
B. `sqrt((17)/(2))`
C. `(15)/(sqrt(34))`
D. `(17)/(2)`

Answer

To find the length of the perpendicular from the point (4, 4) to the line passing through the points (5, 0) and (0, 3), we can follow these steps:

### Step 1: Find the equation of the line
We can use the two-point form of the equation of a line. The formula is:

\[ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1) \]

Here, we have two points: \( A(5, 0) \) and \( B(0, 3) \). So, \( (x_1, y_1) = (5, 0) \) and \( (x_2, y_2) = (0, 3) \).

Substituting the values into the formula:

\[ y - 0 = \frac{3 - 0}{0 - 5} (x - 5) \]

This simplifies to:

\[ y = -\frac{3}{5}(x - 5) \]

Expanding this:

\[ y = -\frac{3}{5}x + 3 \]

Rearranging into standard form \( Ax + By + C = 0 \):

\[ 3x + 5y - 15 = 0 \]

### Step 2: Identify coefficients for the distance formula
From the equation \( 3x + 5y - 15 = 0 \), we identify:
- \( A = 3 \)
- \( B = 5 \)
- \( C = -15 \)

The point from which we want to find the perpendicular distance is \( (x_1, y_1) = (4, 4) \).

### Step 3: Use the distance formula
The formula for the distance \( d \) from a point \( (x_1, y_1) \) to a line \( Ax + By + C = 0 \) is given by:

\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]

Substituting the values:

\[ d = \frac{|3(4) + 5(4) - 15|}{\sqrt{3^2 + 5^2}} \]

Calculating the numerator:

\[ = \frac{|12 + 20 - 15|}{\sqrt{9 + 25}} = \frac{|17|}{\sqrt{34}} = \frac{17}{\sqrt{34}} \]

### Step 4: Simplify the distance
To simplify \( \frac{17}{\sqrt{34}} \):

\[ = \frac{17}{\sqrt{34}} \cdot \frac{\sqrt{34}}{\sqrt{34}} = \frac{17\sqrt{34}}{34} \]

### Final Answer
Thus, the length of the perpendicular from the point (4, 4) to the line is:

\[ \frac{17\sqrt{34}}{34} \]