The perpendicular distance from the point (x_{1}, y_{1}) to the line ax + by + c = 0 is

|(ax_{1} + by_{1} + c) / √[a^{2} + b^{2}]|

**Example 1:**

Find the length of the perpendicular from (2, -3) to the straight line 2x - y + 9 = 0

**Solution :**

The length of the perpendicular from the point (x_{1}, y_{1}) to the line ax + by + c = 0 is

|(ax_{1} + by_{1} + c) / √[a^{2} + b^{2}]|

Substitute.

= |[2(2) - (-3) + 9] / √[2^{2} + (-1)^{2}]|

= |(4 + 3 + 9) / √(4 + 1)|

= |16/√5|

= 16/√5 units

**Example 2 :**

Find the length of the perpendicular from (5,2) to the straight line 3x + 2y - 1 = 0**Solution :**

The length of the perpendicular from the point (x_{1}, y_{1}) to the line ax + by + c = 0 is

|(ax_{1} + by_{1} + c) / √[a^{2} + b^{2}]|

Substitute.

= |[3(5) + 2(2) - 1] / √[3^{2} + 2^{2}]|

= |[15 + 4 - 1] / √[9 + 4]|

= |18/√13|

= 18/√13 units

**Example 3 :**

Find the length of the perpendicular from (0,1) to the straight line x - 3y + 2 = 0**Solution :**

The length of the perpendicular from the point (x_{1}, y_{1}) to the line ax + by + c = 0 is

|(ax_{1} + by_{1} + c) / √[a^{2} + b^{2}]|

Substitute.

= |[1(0) - 3(1) + 2] / √[1^{2} + (-3)^{2}]|

= |[0 - 3 + 2] / √[1 + 9]|

= |-1/√10|

= 1/√10 units

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