Unveiling the Graphs: A Visual Adventure
In this article, we will delve into the fascinating world of mathematics, specifically exploring the concepts of graphs and geometry. We will examine two linear equations and their corresponding graphs on a coordinate plane. Additionally, we will uncover the coordinates of the points where these lines intersect, and calculate the perimeter and area of a triangle formed by these intersecting lines.
Graphing Linear Equations
Let’s start by understanding how to graph linear equations. Consider the equations y = x + 1 and y = -x + 3. To plot these lines on a coordinate plane, we follow these steps:
Find the points where the lines intersect the x-axis and the y-axis.
- For the equation y = x + 1, the line intersects the x-axis at (-1, 0) and the y-axis at (0, 1).
- For the equation y = -x + 3, the line intersects the x-axis at (3, 0) and the y-axis at (0, 3).
Connect the two points obtained for each line to create the graphs.
- The graph of y = x + 1 passes through the points (-1, 0) and (0, 1).
- The graph of y = -x + 3 passes through the points (3, 0) and (0, 3).
Visualize the graphs of these equations in the image below:
Intersecting Points: Solving Equations
Now, let’s determine the coordinates of the points where these two lines intersect.
To find the x-coordinate of the intersection point (C), we solve the equation:
x + 1 = -x + 3
Simplifying the equation, we get:
2x = 2
By dividing both sides by 2, we find:
x = 1
Hence, the x-coordinate of the point (C) is 1.
To find the y-coordinate of the intersection point (C), we substitute the x-coordinate into either of the equations:
y = x + 1
Substituting x = 1, we get:
y = 1 + 1
Therefore, the y-coordinate of the point (C) is 2.
Thus, the coordinates of the intersection point (C) are (1, 2).
Perimeter and Area of the Triangle
Now, let’s calculate the perimeter and area of the triangle formed by the intersection points (A, B, C).
Given that the length of side AB is 4 units, we can apply the Pythagorean theorem to find the lengths of the other sides:
- Side AC: AC = √(2² + 2²) = √(4 + 4) = √8 = 2√2 units
- Side BC: BC = √(2² + 2²) = √(4 + 4) = √8 = 2√2 units
Therefore, the perimeter of triangle ABC is:
AB + BC + AC = 4 + 2√2 + 2√2 = 4 + 4√2 (cm)
Next, let’s calculate the area of triangle ABC using the formula:
Area = (1/2) base height.
In this case, the base is AB, which is 4 units, and the height is 2 units.
Hence, the area of triangle ABC is:
Area = (1/2) 4 2 = 4 (cm²)
That concludes our exploration of the amazing world of graphs and geometry!
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