What is the Foot of the Perpendicular? | Simple Explanation for Class 12

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What is the Foot of the Perpendicular from a Point to a Line?

Grade Level:

Class 12

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Definition

What is it?

The 'foot of the perpendicular' from a point to a line is simply the point on the line that is closest to the given point. Imagine dropping a straight line from the point onto the main line, making a perfect 90-degree angle; where this dropped line touches the main line is the foot of the perpendicular.

Simple Example

Quick Example

Imagine you are standing at a specific spot (the point P) in a big open field. There's a straight road (the line L) running through the field. You want to find the shortest path to reach the road. If you walk straight towards the road, making a perfect 'L' shape with the road, the exact spot where your path meets the road is the foot of the perpendicular. This is the closest point on the road to where you are standing.

Worked Example

Step-by-Step

Let's find the foot of the perpendicular from point P(1, 2) to the line L: y = x + 1.

1. **Understand the Goal:** We need to find a point Q on the line y = x + 1 such that the line segment PQ is perpendicular to y = x + 1.
---2. **Slope of the given line:** The line y = x + 1 has a slope (m1) of 1.
---3. **Slope of the perpendicular line:** For PQ to be perpendicular to L, its slope (m2) must be -1/m1. So, m2 = -1/1 = -1.
---4. **Equation of the perpendicular line:** This line passes through P(1, 2) and has a slope of -1. Using y - y1 = m(x - x1), we get y - 2 = -1(x - 1). This simplifies to y - 2 = -x + 1, or y = -x + 3.
---5. **Find the intersection point:** The foot of the perpendicular (point Q) is where the two lines intersect. We solve the system of equations:
Line L: y = x + 1
Perpendicular line: y = -x + 3
---6. **Solve for x and y:** Substitute the first equation into the second: x + 1 = -x + 3. This gives 2x = 2, so x = 1.
---7. **Find y:** Substitute x = 1 back into y = x + 1: y = 1 + 1 = 2.
---8. **The Foot of the Perpendicular:** The intersection point is Q(1, 2).

**Answer:** The foot of the perpendicular from point P(1, 2) to the line y = x + 1 is (1, 2). (In this specific case, the point P was already on the line L!).

Why It Matters

This concept is crucial in many advanced fields like Computer Graphics for rendering objects accurately or in Robotics for path planning, ensuring a robot takes the shortest, most efficient route. Engineers use it to design optimal structures and routes, making it a foundation for careers in AI, Space Technology, and even urban planning.

Common Mistakes

MISTAKE: Assuming the foot of the perpendicular is always the midpoint of the line segment from the point to some arbitrary point on the line. | CORRECTION: The foot of the perpendicular is *the* unique point on the line that forms a 90-degree angle with the segment connecting it to the given point, ensuring the shortest distance.

MISTAKE: Incorrectly calculating the slope of the perpendicular line. | CORRECTION: Remember that if the original line has slope 'm', the perpendicular line has a slope of '-1/m'. If 'm' is 0, the perpendicular is vertical (undefined slope); if 'm' is undefined, the perpendicular is horizontal (slope 0).

MISTAKE: Stopping after finding the equation of the perpendicular line. | CORRECTION: The goal is to find the *point* on the original line. You must solve the system of equations for both the original line and the perpendicular line to find their intersection point.

Practice Questions

Try It Yourself

QUESTION: What is the foot of the perpendicular from the point (0, 0) to the line x = 5? | ANSWER: (5, 0)

QUESTION: Find the foot of the perpendicular from the point (4, 1) to the line y = 2x. | ANSWER: (2, 4)

QUESTION: A point A is at (3, 5). A line L passes through points B(1, 1) and C(5, 3). Find the coordinates of the foot of the perpendicular from point A to line L. | ANSWER: (4, 2)

MCQ

Quick Quiz

If a point P is on a line L, what is the foot of the perpendicular from P to L?

Any point on the line L

The point P itself

The midpoint of the line segment L

A point outside the line L

The Correct Answer Is:

If the point is already on the line, the shortest distance from the point to the line is zero, and the 'foot' is the point itself. No other point on the line would be closer.

Real World Connection

In the Real World

Think about how your GPS app works! When you're driving your auto-rickshaw on a street (the line) and the GPS shows your current location (the point), the app needs to figure out which exact spot on the road you are closest to. This is precisely finding the foot of the perpendicular. It helps mapping apps like Google Maps or Ola Cabs accurately 'snap' your location onto the road network.

Key Vocabulary

Key Terms

PERPENDICULAR: Forming a 90-degree angle with another line or surface | SLOPE: A measure of the steepness of a line | INTERSECTION: The point where two lines or surfaces cross each other | COORDINATES: A set of numbers used to locate a point in space

What's Next

What to Learn Next

Now that you understand the foot of the perpendicular, you can explore finding the *distance* from a point to a line. This builds directly on this concept, as the shortest distance is always along the perpendicular path. Keep up the great work!