Distance From Point (6, -2) To Line Y = 2x - 4: Explained
Have you ever wondered how to find the shortest distance from a specific point to a line? It's a common problem in mathematics, particularly in coordinate geometry, and it has numerous practical applications in fields like physics, engineering, and computer graphics. In this guide, we'll explore the step-by-step process of calculating the distance from a point to a line, using the example of finding the distance from the point (6, -2) to the line y = 2x - 4. Let's dive in!
Understanding the Fundamentals
Before we jump into the calculations, it's crucial to understand the underlying concepts. The distance from a point to a line is defined as the shortest distance, which is always the perpendicular distance. This means we need to find the length of the line segment that connects the point to the line at a right angle (90 degrees). To do this, we'll use a specific formula derived from the general equation of a line and the coordinates of the point.
The general form of a linear equation is Ax + By + C = 0, and the formula to calculate the distance (d) from a point (x₁, y₁) to this line is given by:
d = |Ax₁ + By₁ + C| / √(A² + B²)
This formula might seem daunting at first, but we'll break it down step by step to make it easy to understand and apply.
Step 1: Convert the Equation to General Form
Our first step is to convert the given equation of the line, y = 2x - 4, into the general form Ax + By + C = 0. To do this, we need to rearrange the equation so that all terms are on one side and the equation is equal to zero.
Subtracting y from both sides of the equation, we get:
0 = 2x - y - 4
Now, we can rewrite this in the general form as:
2x - y - 4 = 0
From this, we can identify the coefficients:
A = 2
B = -1
C = -4
It’s crucial to correctly identify these coefficients as they will be used directly in the distance formula. A simple mistake here can lead to an incorrect answer, so double-check your work!
Step 2: Identify the Point Coordinates
Next, we need to identify the coordinates of the point from which we want to calculate the distance. In this case, the point is given as (6, -2). So, we have:
x₁ = 6
y₁ = -2
These values represent the x and y coordinates of our point, and they will also be plugged into the distance formula.
Step 3: Apply the Distance Formula
Now that we have all the necessary components, we can apply the distance formula:
d = |Ax₁ + By₁ + C| / √(A² + B²)
Substitute the values we found in steps 1 and 2 into the formula:
d = |(2 * 6) + (-1 * -2) + (-4)| / √(2² + (-1)²)
Let's break down the calculation:
Numerator:
(2 * 6) = 12
(-1 * -2) = 2
So, the numerator becomes:
|12 + 2 - 4| = |10|
The absolute value of 10 is simply 10.
Denominator:
2² = 4
(-1)² = 1
So, the denominator becomes:
√(4 + 1) = √5
Therefore, the distance is:
d = 10 / √5
Step 4: Simplify the Result
While 10 / √5 is a correct answer, it's common practice to rationalize the denominator. This means we want to eliminate the square root from the denominator. To do this, we multiply both the numerator and the denominator by √5:
d = (10 / √5) * (√5 / √5)
d = (10√5) / 5
Now, we can simplify further by dividing both the numerator and the denominator by 5:
d = 2√5
So, the exact distance from the point (6, -2) to the line y = 2x - 4 is 2√5 units. If you need a decimal approximation, you can use a calculator to find that 2√5 ≈ 4.47 units.
Alternative Method: Using Perpendicular Lines
Another way to approach this problem is by finding the equation of the line that is perpendicular to the given line and passes through the given point. This method involves a few more steps but provides a deeper understanding of the geometry involved.
Step 1: Find the Slope of the Perpendicular Line
The slope of the given line, y = 2x - 4, is 2. The slope of a line perpendicular to this line is the negative reciprocal of 2, which is -1/2.
Step 2: Find the Equation of the Perpendicular Line
We know the perpendicular line has a slope of -1/2 and passes through the point (6, -2). We can use the point-slope form of a line equation:
y - y₁ = m(x - x₁)
Where m is the slope and (x₁, y₁) is the point. Plugging in our values:
y - (-2) = (-1/2)(x - 6)
y + 2 = (-1/2)x + 3
Subtracting 2 from both sides, we get the equation of the perpendicular line:
y = (-1/2)x + 1
Step 3: Find the Intersection Point
To find the point where the two lines intersect, we set their y-values equal to each other:
2x - 4 = (-1/2)x + 1
Multiplying both sides by 2 to eliminate the fraction:
4x - 8 = -x + 2
Adding x to both sides:
5x - 8 = 2
Adding 8 to both sides:
5x = 10
Dividing by 5:
x = 2
Now, substitute x = 2 into either equation to find the y-coordinate. Let’s use y = 2x - 4:
y = 2(2) - 4
y = 4 - 4
y = 0
So, the intersection point is (2, 0).
Step 4: Calculate the Distance
Now that we have the point (6, -2) and the intersection point (2, 0), we can use the distance formula to find the distance between these two points:
d = √((x₂ - x₁)² + (y₂ - y₁)²)
d = √((2 - 6)² + (0 - (-2))²)
d = √((-4)² + (2)²)
d = √(16 + 4)
d = √20
d = √(4 * 5)
d = 2√5
Again, we find that the distance is 2√5 units, confirming our previous result.
Common Mistakes to Avoid
When calculating the distance from a point to a line, there are a few common mistakes that you should be aware of:
- Incorrectly Identifying Coefficients: Make sure you correctly identify the values of A, B, and C when converting the equation to general form. A mistake here will propagate through the rest of the calculation.
- Forgetting the Absolute Value: The distance formula includes an absolute value in the numerator. Failing to use the absolute value can result in a negative distance, which doesn't make sense in this context.
- Not Rationalizing the Denominator: While not strictly a mistake, it's good mathematical practice to rationalize the denominator. This makes the answer cleaner and easier to compare with other results.
- Misapplying the Distance Formula: Ensure you're using the correct distance formula for a point to a line, not the distance formula for two points. They are different!
Practical Applications
The ability to calculate the distance from a point to a line has numerous practical applications. Here are a few examples:
- Navigation: In navigation systems, this calculation is used to determine how far a vehicle or vessel is from a planned route.
- Computer Graphics: In computer graphics, it's used for collision detection and determining the closest point on a surface to a given point.
- Physics: In physics, it can be used to calculate the shortest distance a particle needs to travel to reach a certain trajectory.
- Engineering: In engineering, it's used in structural analysis to determine the distance from a load to a support beam.
Conclusion
Calculating the distance from a point to a line is a fundamental concept in coordinate geometry with wide-ranging applications. By understanding the formula and following the steps outlined in this guide, you can confidently solve these types of problems. Whether you choose to use the direct distance formula or the perpendicular line method, the key is to be methodical and pay attention to detail. Remember to double-check your work and rationalize the denominator for the most accurate and presentable answer. So next time you encounter a problem asking for the distance from a point to a line, you’ll be well-equipped to tackle it!
For further learning and practice on related topics, you might find the resources available at Khan Academy particularly helpful.
