Finding The Equation Of A Perpendicular Line A Step By Step Guide

by ADMIN 66 views

Hey everyone! Let's dive into a cool math problem today that involves finding the equation of a line perpendicular to a given line and passing through a specific point. It's a classic problem that combines concepts of slopes and line equations. So, grab your pencils, and let's get started!

Understanding the Problem

The equation of the line QR is given as x + 2y = 2. Our mission, should we choose to accept it (and we do!), is to find the equation of a line that is perpendicular to QR and contains the point (5, 6). To solve this, we'll need to understand a few key concepts:

  1. Slope-intercept form: This is the y = mx + b format, where m is the slope and b is the y-intercept.
  2. Perpendicular lines: Perpendicular lines have slopes that are negative reciprocals of each other. If one line has a slope of m, a line perpendicular to it has a slope of -1/m.
  3. Point-slope form: This is y - y₁ = m( x - x₁), which is super useful when you know a point on the line (x₁, y₁) and the slope (m).

Let's break down each of these concepts a bit more to make sure we're all on the same page. First up, the slope-intercept form. This form is awesome because it immediately tells us two important things about a line: its slope and where it crosses the y-axis. The slope (m) tells us how steep the line is and whether it's going uphill or downhill as you move from left to right. The y-intercept (b) is the point where the line intersects the y-axis. Knowing these two pieces of information makes it super easy to visualize and graph the line.

Now, let's talk about perpendicular lines. This is where the concept of negative reciprocals comes into play. Imagine two lines crossing each other at a perfect right angle (90 degrees). That's what it means for lines to be perpendicular. The relationship between their slopes is key. If one line has a slope of, say, 2, then a line perpendicular to it will have a slope of -1/2. Notice the flip (reciprocal) and the change in sign (negative). This relationship ensures the lines intersect at a right angle. Understanding this negative reciprocal relationship is crucial for solving problems like the one we're tackling today.

Finally, the point-slope form. This form is like the Swiss Army knife of line equations. It's incredibly versatile because it allows you to write the equation of a line if you know just one point on the line and the slope. Think about it: if you know a specific location the line passes through and how steep it is, you can define the entire line. The point-slope form y - y₁ = m( x - x₁) directly uses this information. Here, (x₁, y₁) is the known point, and m is the slope. This form is particularly useful when you need to find the equation of a line given a point and a slope, which is exactly what we're going to do in our problem. It's a stepping stone to getting the equation into slope-intercept form, which is our final goal.

Step-by-Step Solution

Step 1: Find the Slope of Line QR

The first step in solving our problem is to figure out the slope of the given line, x + 2y = 2. To do this, we need to rearrange the equation into slope-intercept form (y = mx + b). This will make the slope (m) readily apparent.

Let's rewrite the equation x + 2y = 2. Our goal is to isolate y on one side of the equation. First, we subtract x from both sides:

2y = -x + 2

Next, we divide both sides by 2 to get y by itself:

y = (-1/2)x + 1

Now we have the equation in slope-intercept form! Comparing this to y = mx + b, we can clearly see that the slope (m) of line QR is -1/2. This is a crucial piece of information because we need to find the slope of a line perpendicular to this one.

Step 2: Determine the Slope of the Perpendicular Line

Remember, perpendicular lines have slopes that are negative reciprocals of each other. This means we need to flip the slope of line QR (which is -1/2) and change its sign. The negative reciprocal of -1/2 is:

-(-2/1) = 2

So, the slope of any line perpendicular to QR is 2. Now we know the slope of the line we're trying to find! We're halfway there.

Step 3: Use Point-Slope Form

We now know the slope of our perpendicular line (which is 2) and a point it passes through: (5, 6). This is the perfect setup for using the point-slope form of a line equation, which, as we discussed earlier, is:

y - y₁ = m(x - x₁)

Here, m is the slope, and (x₁, y₁) is the point. We can plug in our values:

y - 6 = 2(x - 5)

This equation represents the line we want, but it's not in slope-intercept form yet. We need to do a little bit of algebra to get it into the familiar y = mx + b format.

Step 4: Convert to Slope-Intercept Form

Let's take our equation from the point-slope form and transform it into slope-intercept form. We have:

y - 6 = 2(x - 5)

First, distribute the 2 on the right side of the equation:

y - 6 = 2x - 10

Now, to isolate y, add 6 to both sides:

y = 2x - 10 + 6

Simplify:

y = 2x - 4

Ta-da! We have our equation in slope-intercept form. We can see that the slope is 2 (which we already knew) and the y-intercept is -4.

The Answer

The equation of the line perpendicular to QR and containing the point (5, 6) in slope-intercept form is:

y = 2x - 4

This corresponds to option B in the given choices. Fantastic!

Key Takeaways

  • Understanding slope-intercept form (y = mx + b) is crucial for identifying the slope and y-intercept of a line.
  • Perpendicular lines have slopes that are negative reciprocals of each other.
  • The point-slope form (y - y₁ = m(x - x₁)) is a powerful tool for finding the equation of a line when you know a point and the slope.
  • Converting between point-slope form and slope-intercept form is a valuable algebraic skill.

Practice Makes Perfect

To really master these concepts, try solving similar problems. Change the given line equation or the point, and see if you can still find the equation of the perpendicular line. The more you practice, the more comfortable you'll become with these types of problems. You've got this!

I hope this step-by-step explanation helped you understand how to solve this type of problem. Keep practicing, and you'll be a pro in no time! If you have any questions, feel free to ask. Happy problem-solving, guys!