Select The Correct Answer.When Graphed, The Three Lines { Y = -x + 2 $}$, { Y = 2x - 1 $}$, And { Y = X - 2 $}$ Intersect In Such A Way That They Form A Triangle. What Are The Coordinates Of The Three Vertices Of This
Introduction
When graphing three lines, it's not uncommon for them to intersect in such a way that they form a triangle. In this case, we're given three lines: { y = -x + 2 $}$, { y = 2x - 1 $}$, and { y = x - 2 $}$. Our goal is to find the coordinates of the three vertices of the triangle formed by the intersection of these three lines.
Understanding the Problem
To solve this problem, we need to find the points of intersection between each pair of lines. This will give us the coordinates of the vertices of the triangle. We can start by finding the intersection points between each pair of lines.
Finding the Intersection Points
To find the intersection points, we need to set the equations of the lines equal to each other and solve for the value of x. We can then substitute this value of x into one of the original equations to find the corresponding value of y.
Intersection Point 1: Lines 1 and 2
Let's start by finding the intersection point between lines 1 and 2. We can set the equations equal to each other and solve for x:
{ -x + 2 = 2x - 1 $}$
Simplifying the equation, we get:
{ 3x = 3 $}$
Dividing both sides by 3, we get:
{ x = 1 $}$
Now that we have the value of x, we can substitute it into one of the original equations to find the corresponding value of y. Let's use the equation of line 1:
{ y = -x + 2 $}$
Substituting x = 1, we get:
{ y = -1 + 2 $}$
Simplifying the equation, we get:
{ y = 1 $}$
So, the intersection point between lines 1 and 2 is (1, 1).
Intersection Point 2: Lines 2 and 3
Next, let's find the intersection point between lines 2 and 3. We can set the equations equal to each other and solve for x:
{ 2x - 1 = x - 2 $}$
Simplifying the equation, we get:
{ x = -1 $}$
Now that we have the value of x, we can substitute it into one of the original equations to find the corresponding value of y. Let's use the equation of line 2:
{ y = 2x - 1 $}$
Substituting x = -1, we get:
{ y = 2(-1) - 1 $}$
Simplifying the equation, we get:
{ y = -3 $}$
So, the intersection point between lines 2 and 3 is (-1, -3).
Intersection Point 3: Lines 1 and 3
Finally, let's find the intersection point between lines 1 and 3. We can set the equations equal to each other and solve for x:
{ -x + 2 = x - 2 $}$
Simplifying the equation, we get:
{ 2x = 4 $}$
Dividing both sides by 2, we get:
{ x = 2 $}$
Now that we have the value of x, we can substitute it into one of the original equations to find the corresponding value of y. Let's use the equation of line 1:
{ y = -x + 2 $}$
Substituting x = 2, we get:
{ y = -2 + 2 $}$
Simplifying the equation, we get:
{ y = 0 $}$
So, the intersection point between lines 1 and 3 is (2, 0).
Conclusion
In conclusion, we have found the coordinates of the three vertices of the triangle formed by the intersection of the three lines. The vertices are (1, 1), (-1, -3), and (2, 0).
Final Answer
The final answer is:
Introduction
In our previous article, we explored how to find the coordinates of the three vertices of a triangle formed by the intersection of three lines. We used the equations of the lines to find the points of intersection between each pair of lines. In this article, we'll answer some common questions related to this topic.
Q: What are the three lines used to form the triangle?
A: The three lines used to form the triangle are:
{ y = -x + 2 $}{$ y = 2x - 1 $}{$ y = x - 2 $}$
Q: How do you find the intersection points between each pair of lines?
A: To find the intersection points, we need to set the equations of the lines equal to each other and solve for the value of x. We can then substitute this value of x into one of the original equations to find the corresponding value of y.
Q: What is the process for finding the intersection points?
A: The process for finding the intersection points involves the following steps:
- Set the equations of the lines equal to each other.
- Solve for the value of x.
- Substitute the value of x into one of the original equations to find the corresponding value of y.
Q: How do you determine the coordinates of the vertices of the triangle?
A: The coordinates of the vertices of the triangle are found by identifying the points of intersection between each pair of lines. These points are the vertices of the triangle.
Q: What are the coordinates of the three vertices of the triangle?
A: The coordinates of the three vertices of the triangle are:
{ (1, 1) $}{$ (-1, -3) $}{$ (2, 0) $}$
Q: Why is it important to find the intersection points of the lines?
A: Finding the intersection points of the lines is important because it allows us to determine the coordinates of the vertices of the triangle. This information can be used to graph the triangle and understand its properties.
Q: Can you provide an example of how to use the intersection points to graph the triangle?
A: Yes, here's an example of how to use the intersection points to graph the triangle:
- Plot the points of intersection on a coordinate plane.
- Draw a line through each pair of points to form the sides of the triangle.
- Label the vertices of the triangle with their coordinates.
Q: What are some real-world applications of finding the intersection points of lines?
A: Finding the intersection points of lines has many real-world applications, including:
- Graphing functions and understanding their behavior.
- Solving systems of equations and finding the solution set.
- Determining the coordinates of points in a coordinate plane.
Conclusion
In conclusion, finding the intersection points of lines is an important concept in mathematics that has many real-world applications. By understanding how to find the intersection points, we can graph functions, solve systems of equations, and determine the coordinates of points in a coordinate plane.
Final Answer
The final answer is: