{ \overleftrightarrow{AB}$}$ And { \overleftrightarrow{BC}$}$ Form A Right Angle At Point { B$}$. If { A=(-3,-1)$}$ And { B=(4,4)$}$, What Is The Equation Of { \overleftrightarrow{BC}$}$?A.
Introduction
In geometry, the equation of a line can be found using various methods, including the slope-intercept form, point-slope form, and the two-point form. In this article, we will focus on finding the equation of a line given two points that form a right angle at one of the points. We will use the two-point form to find the equation of the line.
The Two-Point Form
The two-point form of a line is given by the equation:
y - y1 = m(x - x1)
where (x1, y1) and (x2, y2) are two points on the line, and m is the slope of the line.
Finding the Slope
To find the slope of the line, we can use the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are two points on the line.
Given Points
We are given two points: A = (-3, -1) and B = (4, 4). We need to find the equation of the line BC.
Finding the Slope of the Line BC
To find the slope of the line BC, we can use the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) = (4, 4) and (x2, y2) = (-3, -1).
m = (4 - (-1)) / (4 - (-3)) m = (4 + 1) / (4 + 3) m = 5 / 7
Finding the Equation of the Line BC
Now that we have the slope of the line BC, we can use the point-slope form to find the equation of the line:
y - y1 = m(x - x1)
where (x1, y1) = (4, 4) and m = 5/7.
y - 4 = (5/7)(x - 4)
Simplifying the Equation
To simplify the equation, we can multiply both sides by 7 to eliminate the fraction:
7(y - 4) = 5(x - 4)
Expanding the equation, we get:
7y - 28 = 5x - 20
Adding 28 to both sides, we get:
7y = 5x + 8
Conclusion
In this article, we found the equation of the line BC given two points A = (-3, -1) and B = (4, 4) that form a right angle at point B. We used the two-point form to find the equation of the line. The equation of the line BC is 7y = 5x + 8.
Example Problems
- Find the equation of the line AB given two points A = (2, 3) and B = (5, 6).
- Find the equation of the line CD given two points C = (-2, 1) and D = (3, 4).
Step-by-Step Solutions
- Find the slope of the line AB:
m = (y2 - y1) / (x2 - x1) m = (6 - 3) / (5 - 2) m = 3 / 3 m = 1
- Find the equation of the line AB:
y - y1 = m(x - x1) y - 3 = 1(x - 2) y - 3 = x - 2 y = x - 1
- Find the slope of the line CD:
m = (y2 - y1) / (x2 - x1) m = (4 - 1) / (3 - (-2)) m = 3 / 5
- Find the equation of the line CD:
y - y1 = m(x - x1) y - 1 = (3/5)(x - (-2)) y - 1 = (3/5)(x + 2) y - 1 = (3/5)x + 6/5 y = (3/5)x + 31/5
Final Answer
Introduction
In our previous article, we discussed how to find the equation of a line given two points that form a right angle at one of the points. We used the two-point form to find the equation of the line. In this article, we will answer some frequently asked questions related to finding the equation of a line given two points.
Q: What is the two-point form of a line?
A: The two-point form of a line is given by the equation:
y - y1 = m(x - x1)
where (x1, y1) and (x2, y2) are two points on the line, and m is the slope of the line.
Q: How do I find the slope of a line given two points?
A: To find the slope of a line given two points, you can use the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are two points on the line.
Q: What if the two points are not on the same horizontal or vertical line?
A: If the two points are not on the same horizontal or vertical line, you can still find the slope of the line using the formula:
m = (y2 - y1) / (x2 - x1)
Q: How do I find the equation of a line given two points that form a right angle at one of the points?
A: To find the equation of a line given two points that form a right angle at one of the points, you can use the two-point form. First, find the slope of the line using the formula:
m = (y2 - y1) / (x2 - x1)
Then, use the point-slope form to find the equation of the line:
y - y1 = m(x - x1)
Q: What if I have three points and I want to find the equation of the line that passes through them?
A: If you have three points and you want to find the equation of the line that passes through them, you can use the following steps:
- Find the slope of the line using the formula:
m = (y2 - y1) / (x2 - x1)
- Use the point-slope form to find the equation of the line:
y - y1 = m(x - x1)
- Repeat steps 1 and 2 for the other two points.
Q: Can I use the two-point form to find the equation of a line that is not a straight line?
A: No, the two-point form can only be used to find the equation of a straight line. If you have a curve or a non-linear line, you will need to use a different method to find its equation.
Q: How do I know if the two points are on the same line?
A: To determine if the two points are on the same line, you can use the following steps:
- Find the slope of the line using the formula:
m = (y2 - y1) / (x2 - x1)
- If the slope is the same for both points, then the points are on the same line.
Q: Can I use the two-point form to find the equation of a line that is parallel to another line?
A: Yes, you can use the two-point form to find the equation of a line that is parallel to another line. To do this, you will need to find the slope of the original line and then use the point-slope form to find the equation of the new line.
Conclusion
In this article, we answered some frequently asked questions related to finding the equation of a line given two points. We discussed the two-point form, how to find the slope of a line given two points, and how to find the equation of a line given two points that form a right angle at one of the points. We also discussed how to find the equation of a line given three points and how to determine if two points are on the same line.