{ \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 determined using various methods, including the slope-intercept form, point-slope form, and two-point form. In this article, we will focus on finding the equation of a line given two points. Specifically, we will use the two-point form to find the equation of line {\overleftrightarrow{BC}$}$ given points {A=(-3,-1)$}$ and {B=(4,4)$}$.
The Two-Point Form
The two-point form of a line is given by the equation:
{y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)$}$
where {(x_1,y_1)$}$ and {(x_2,y_2)$}$ are the coordinates of the two points.
Step 1: Identify the Coordinates of the Two Points
We are given the coordinates of points {A=(-3,-1)$}$ and {B=(4,4)$}$. We can identify these coordinates as {(x_1,y_1)=(-3,-1)$}$ and {(x_2,y_2)=(4,4)$}$.
Step 2: Plug in the Values into the Two-Point Form
We can now plug in the values of {(x_1,y_1)$}$ and {(x_2,y_2)$}$ into the two-point form:
{y-(-1)=\frac{4-(-1)}{4-(-3)}(x-(-3))$}$
Step 3: Simplify the Equation
We can simplify the equation by evaluating the expressions:
{y+1=\frac{5}{7}(x+3)$}$
Step 4: Write the Equation in Slope-Intercept Form
We can rewrite the equation in slope-intercept form by isolating {y$}$:
{y=\frac{5}{7}(x+3)-1$}$
{y=\frac{5}{7}x+\frac{15}{7}-1$}$
{y=\frac{5}{7}x+\frac{8}{7}$}$
Conclusion
In this article, we used the two-point form to find the equation of line {\overleftrightarrow{BC}$}$ given points {A=(-3,-1)$}$ and {B=(4,4)$}$. We identified the coordinates of the two points, plugged in the values into the two-point form, simplified the equation, and finally wrote the equation in slope-intercept form. The equation of line {\overleftrightarrow{BC}$}$ is {y=\frac{5}{7}x+\frac{8}{7}$}$.
Example Use Case
Suppose we want to find the equation of line {\overleftrightarrow{BC}$}$ given points {A=(2,3)$}$ and {B=(5,6)$}$. We can use the two-point form to find the equation of the line:
{y-3=\frac{6-3}{5-2}(x-2)$}$
{y-3=\frac{3}{3}(x-2)$}$
{y-3=1(x-2)$}$
{y-3=x-2$}$
{y=x+1$}$
Tips and Tricks
- When using the two-point form, make sure to identify the coordinates of the two points correctly.
- When plugging in the values into the two-point form, make sure to simplify the expressions.
- When rewriting the equation in slope-intercept form, make sure to isolate {y$}$.
Common Mistakes
- Failing to identify the coordinates of the two points correctly.
- Failing to simplify the expressions when plugging in the values into the two-point form.
- Failing to isolate {y$}$ when rewriting the equation in slope-intercept form.
Conclusion
Introduction
In our previous article, we discussed how to find the equation of a line given two points using the two-point form. 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-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)$}$
where {(x_1,y_1)$}$ and {(x_2,y_2)$}$ are the coordinates of the two points.
Q: How do I find the equation of a line given two points?
A: To find the equation of a line given two points, you can use the two-point form. First, identify the coordinates of the two points. Then, plug in the values into the two-point form and simplify the equation. Finally, rewrite the equation in slope-intercept form.
Q: What is the slope-intercept form of a line?
A: The slope-intercept form of a line is given by the equation:
{y=mx+b$}$
where {m$}$ is the slope and {b$}$ is the y-intercept.
Q: How do I convert the two-point form to the slope-intercept form?
A: To convert the two-point form to the slope-intercept form, you can isolate {y$}$ by adding {y_1$}$ to both sides of the equation and then dividing both sides by the coefficient of {x$}$.
Q: What is the y-intercept of a line?
A: The y-intercept of a line is the point where the line intersects the y-axis. It is the value of {y$}$ when {x=0$}$.
Q: How do I find the y-intercept of a line?
A: To find the y-intercept of a line, you can set {x=0$}$ in the equation of the line and solve for {y$}$.
Q: What is the slope of a line?
A: The slope of a line is a measure of how steep the line is. It is the ratio of the vertical change to the horizontal change.
Q: How do I find the slope of a line?
A: To find the slope of a line, you can use the formula:
{m=\frac{y_2-y_1}{x_2-x_1}$}$
where {(x_1,y_1)$}$ and {(x_2,y_2)$}$ are the coordinates of two points on the line.
Q: What is the difference between the slope and the y-intercept?
A: The slope is a measure of how steep the line is, while the y-intercept is the point where the line intersects the y-axis.
Q: How do I use the two-point form to find the equation of a line?
A: To use the two-point form to find the equation of a line, you can follow these steps:
- Identify the coordinates of the two points.
- Plug in the values into the two-point form.
- Simplify the equation.
- Rewrite the equation in slope-intercept form.
Conclusion
In conclusion, finding the equation of a line given two points is a fundamental concept in geometry. The two-point form is a useful tool for finding the equation of a line, and it can be used to solve a variety of problems. By following the steps outlined in this article, you can find the equation of a line given two points.
Example Use Case
Suppose we want to find the equation of a line given two points {A=(2,3)$}$ and {B=(5,6)$}$. We can use the two-point form to find the equation of the line:
{y-3=\frac{6-3}{5-2}(x-2)$}$
{y-3=\frac{3}{3}(x-2)$}$
{y-3=1(x-2)$}$
{y-3=x-2$}$
{y=x+1$}$
Tips and Tricks
- When using the two-point form, make sure to identify the coordinates of the two points correctly.
- When plugging in the values into the two-point form, make sure to simplify the expressions.
- When rewriting the equation in slope-intercept form, make sure to isolate {y$}$.
Common Mistakes
- Failing to identify the coordinates of the two points correctly.
- Failing to simplify the expressions when plugging in the values into the two-point form.
- Failing to isolate {y$}$ when rewriting the equation in slope-intercept form.