Penerapan Teorema Pythagoras dalam Menghitung Jarak Titik ke Garis

The Pythagorean theorem, a fundamental concept in geometry, finds its application in various real-world scenarios, including calculating the distance between a point and a line. This theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides, provides a powerful tool for determining distances in geometric figures. This article delves into the application of the Pythagorean theorem in calculating the distance between a point and a line, exploring the underlying principles and providing practical examples to illustrate the process.

Understanding the Concept

The distance between a point and a line is defined as the shortest distance between the point and any point on the line. This shortest distance is always perpendicular to the line. To calculate this distance using the Pythagorean theorem, we need to construct a right-angled triangle where the distance between the point and the line is the hypotenuse, and the other two sides are the perpendicular distance from the point to the line and the distance along the line from the foot of the perpendicular to the point where the line intersects the perpendicular.

Applying the Theorem

Let's consider a point P with coordinates (x1, y1) and a line with the equation ax + by + c = 0. To find the distance between point P and the line, we follow these steps:

1. Find the equation of the perpendicular line: The perpendicular line passing through point P will have a slope that is the negative reciprocal of the slope of the given line. The slope of the given line is -a/b, so the slope of the perpendicular line is b/a. Using the point-slope form of a line, the equation of the perpendicular line is y - y1 = (b/a)(x - x1).

2. Find the point of intersection: Solve the equations of the given line and the perpendicular line simultaneously to find the point of intersection, which is the foot of the perpendicular.

3. Calculate the distance: Once we have the coordinates of the point of intersection and the coordinates of point P, we can use the distance formula to calculate the distance between them. This distance represents the shortest distance between point P and the line.

Example

Let's illustrate this with an example. Suppose we have a point P(2, 3) and a line with the equation 2x + 3y - 6 = 0.

1. Equation of the perpendicular line: The slope of the given line is -2/3, so the slope of the perpendicular line is 3/2. Using the point-slope form, the equation of the perpendicular line is y - 3 = (3/2)(x - 2).

2. Point of intersection: Solving the equations 2x + 3y - 6 = 0 and y - 3 = (3/2)(x - 2) simultaneously, we get the point of intersection as (1, 2).

3. Distance: Using the distance formula, the distance between P(2, 3) and (1, 2) is √((2 - 1)² + (3 - 2)²) = √2.

Therefore, the distance between point P(2, 3) and the line 2x + 3y - 6 = 0 is √2 units.

Conclusion

The Pythagorean theorem provides a powerful tool for calculating the distance between a point and a line. By constructing a right-angled triangle and applying the theorem, we can determine the shortest distance between the point and any point on the line. This method finds applications in various fields, including geometry, engineering, and physics, where determining distances is crucial for solving problems and understanding spatial relationships.