What are the properties of perpendicular lines and how can they be used to find the equation of a line passing through a specific point?
Perpendicular lines have several properties that can be used to find the equation of a line passing through a specific point.
Firstly, two lines are perpendicular if the product of their slopes is equal to -1. This means that if the slope of one line is m1, and the slope of another line is m2, then m1 * m2 = -1.
To find the equation of a line passing through a specific point using perpendicular lines, follow these steps:
1. Start with the equation of the line that passes through the specific point, let's call it (x1, y1), and has a given slope, let's call it m.
- The equation can be written in the form: y - y1 = m(x - x1).
2. Find the negative reciprocal of the given slope to get the slope of the perpendicular line.
- For example, if the given slope is m, the slope of the perpendicular line would be -1/m.
3. Use the slope-intercept form of the equation (y = mx + b) to represent the perpendicular line. Since we now know the slope (-1/m), we need to find the y-intercept (b).
4. Substitute the known point (x1, y1) into the equation to solve for the y-intercept (b).
- Plug in the values of x1 and y1 into the equation from step 3, and solve for b.
5. Write the equation of the perpendicular line in slope-intercept form, using the found slope and y-intercept. This equation will have the form: y = (-1/m)x + b.
Now you have the equation of the line passing through the given point that is perpendicular to the original line.
Perpendicular lines have several important properties:
1. The slopes of perpendicular lines are negative reciprocals of each other. This means that if the slope of one line is m, then the slope of a line perpendicular to it is (-1/m). For example, if one line has a slope of 2, a line perpendicular to it will have a slope of (-1/2).
2. Perpendicular lines intersect at a 90-degree angle. This means that if you have two lines, and you know that they are perpendicular, you can find the point where they intersect by solving their equations simultaneously.
To find the equation of a line passing through a specific point, you can use the concept of perpendicular lines in the following steps:
Step 1: Determine the given point and the slope of the line passing through it.
Step 2: Find the negative reciprocal of the slope. This will give you the slope of the line perpendicular to it.
Step 3: Use the point-slope form of a linear equation, y - y1 = m(x - x1), where (x1, y1) is the given point and m is the negative reciprocal slope.
Step 4: Simplify and rearrange the equation to the desired form. This might involve distributing the negative reciprocal slope and combining like terms.
Step 5: Verify your equation by plugging in the coordinates of the given point to confirm that they satisfy the equation.
By following these steps, you can find the equation of a line passing through a specific point using the concept of perpendicular lines.