Calculate the length of the perpendicular from (7, 0) to the straight line `5x+12y-9=0` and show that it is twice the length of the perpendicular from (2, 1).

Answer

To solve the problem, we will calculate the perpendicular distances from the given points to the line using the formula for the distance from a point to a line. The line is given by the equation \(5x + 12y - 9 = 0\).

### Step 1: Identify the coefficients of the line equation
The line equation can be written in the form \(Ax + By + C = 0\), where:
- \(A = 5\)
- \(B = 12\)
- \(C = -9\)

### Step 2: Calculate the distance from point (7, 0) to the line
We will use the formula for the distance \(d\) from a point \((x_1, y_1)\) to the line \(Ax + By + C = 0\):
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
For point \(A(7, 0)\):
- \(x_1 = 7\)
- \(y_1 = 0\)

Substituting the values into the formula:
\[ d_1 = \frac{|5(7) + 12(0) - 9|}{\sqrt{5^2 + 12^2}} = \frac{|35 + 0 - 9|}{\sqrt{25 + 144}} = \frac{|26|}{\sqrt{169}} = \frac{26}{13} = 2 \]

### Step 3: Calculate the distance from point (2, 1) to the line
Now, we calculate the distance from point \(B(2, 1)\):
- \(x_1 = 2\)
- \(y_1 = 1\)

Substituting the values into the formula:
\[ d_2 = \frac{|5(2) + 12(1) - 9|}{\sqrt{5^2 + 12^2}} = \frac{|10 + 12 - 9|}{\sqrt{25 + 144}} = \frac{|13|}{\sqrt{169}} = \frac{13}{13} = 1 \]

### Step 4: Compare the distances
We found:
- \(d_1 = 2\) (distance from point (7, 0))
- \(d_2 = 1\) (distance from point (2, 1))

Now, we can see that:
\[ d_1 = 2 \times d_2 \]
Thus, the distance from point (7, 0) is indeed twice the distance from point (2, 1).

### Conclusion
The length of the perpendicular from (7, 0) to the line \(5x + 12y - 9 = 0\) is 2 units, and it is twice the length of the perpendicular from (2, 1), which is 1 unit.