What will be the length of the perpendicular drawn from the point (4, 5) upon the straight line 3x + 4y = 10?

Options

A. `(12)/(5)`
B. `(32)/(5)`
C. `(22)/(5)`
D. `(42)/(5)`

Answer

To find the length of the perpendicular drawn from the point (4, 5) to the straight line given by the equation \(3x + 4y = 10\), we can use the formula for the perpendicular distance from a point to a line. The formula is:

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

where \(Ax + By + C = 0\) is the standard form of the line equation, and \((x_1, y_1)\) is the point from which we are measuring the distance.

### Step 1: Rewrite the line equation in standard form
The given line equation is:

\[ 3x + 4y = 10 \]

We can rewrite it in the standard form \(Ax + By + C = 0\):

\[ 3x + 4y - 10 = 0 \]

Here, \(A = 3\), \(B = 4\), and \(C = -10\).

### Step 2: Identify the coordinates of the point
The point from which we want to find the perpendicular distance is:

\[ (x_1, y_1) = (4, 5) \]

### Step 3: Substitute values into the formula
Now we substitute \(A\), \(B\), \(C\), \(x_1\), and \(y_1\) into the formula:

\[ d = \frac{|3(4) + 4(5) - 10|}{\sqrt{3^2 + 4^2}} \]

### Step 4: Calculate the numerator
Calculating the numerator:

\[ 3(4) + 4(5) - 10 = 12 + 20 - 10 = 22 \]

Thus, the absolute value is:

\[ |22| = 22 \]

### Step 5: Calculate the denominator
Now, calculate the denominator:

\[ \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 6: Calculate the distance
Now we can find the distance \(d\):

\[ d = \frac{22}{5} = 4.4 \]

### Final Answer
The length of the perpendicular drawn from the point (4, 5) to the line \(3x + 4y = 10\) is \(4.4\).

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