Learn how to find the distance from a point to a line using the formula for the distance between a point and a line in coordinate geometry.

distance from a point to a line

To find the distance from a point to a line, you can use the formula for the distance between a point and a line in coordinate geometry

To find the distance from a point to a line, you can use the formula for the distance between a point and a line in coordinate geometry. The formula is derived from the concept of perpendicular distance.

Let’s assume we have a point P(x₁, y₁) and a line in the form of Ax + By + C = 0. The distance, d, from the point to the line is given by:

d = |Ax₁ + By₁ + C| / √(A² + B²)

Here’s how to apply the formula step by step:
1. Identify the given point and the line.

2. Determine the values of A, B, and C from the equation of the line. For example, if the equation of the line is 3x + 4y – 5 = 0, then A = 3, B = 4, and C = -5.

3. Substitute the values of x₁ and y₁ from the coordinates of the point into the formula.

4. Calculate the numerator (Ax₁ + By₁ + C) and the denominator (√(A² + B²)) separately.

5. Take the absolute value of the numerator (|Ax₁ + By₁ + C|) to ensure a positive distance.

6. Finally, divide the absolute value of the numerator by the denominator to obtain the distance.

For example, let’s say we have a point P(2, 4) and a line with the equation 2x – 3y + 6 = 0. To find the distance from the point to the line:

A = 2, B = -3, C = 6
x₁ = 2, y₁ = 4

Substituting the values into the formula:

d = |2(2) – 3(4) + 6| / √(2² + (-3)²)
= |-4 – 12 + 6| / √(4 + 9)
= |-10| / √13
= 10 / √13

Therefore, the distance from the point P(2, 4) to the line 2x – 3y + 6 = 0 is 10 / √13, which is an irrational number.

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