Solving a Rectangle's Sides and Area Using Linear Equations
In this article, we'll learn how to determine the sides and area of a rectangle when given certain linear equations. We will use the equation of one side of the rectangle, determine the other sides, and finally calculate the area of the rectangle.
Given Equations and Information
The problem involves solving for the sides and area of a rectangle where one side, represented by the equation (4x 7y - 5 0), and two vertices ((-3, 1)) and ((11, 1)) are provided.
Step 1: Recognizing the Given Side
Given: The equation of one side of the rectangle is [4x 7y - 5 0]. This implies that ((-3, 1)) lies on this side. We need to find the equations of the other sides, including the other two vertices.
Step 2: Finding the Perpendicular Side
The slope of the given side is calculated as (-frac{4}{7}). Therefore, the slope of any line perpendicular to it would be (frac{7}{4}). Using this information:
Equation of the perpendicular side through the point ((-3, 1)):[y - 1 frac{7}{4}(x 3) Rightarrow 4y - 4 7x 21 Rightarrow 4x - 7y 25 0]
Step 3: Identifying the Other Vertices
We have one vertex on the line (4x - 7y 25 0). Now, let's find the equation of the side passing through ((11, 1)).
Equation of the side passing through ((11, 1)) with a perpendicular slope:[y - 1 frac{7}{4}(x - 11) Rightarrow 4(y - 1) 7(x - 11) Rightarrow 4x - 7y 77 0 Rightarrow 4x - 7y - 77 0]
Step 4: Calculating the Coordinates of the Vertices
We now have two sides: (4x 7y - 5 0) and (4x - 7y - 77 0). The constant term in the perpendicular equation is (-77), indicating the other vertex to be ((11, 1)).
Since ((11, 1)) also lies on another side, we arrange the side's equation accordingly. The side through ((-3, 1)) will be (7x - 4y 25 0).
Step 5: Calculating the Length of the Diagonal and Area
The length of the diagonal is calculated as [11 - (-3) 14]. Using the distance formula, the length of the diagonal is 14 units.
The area of the rectangle can be found using the formula (frac{1}{2} times text{diagonal}^2 frac{1}{2} times 14^2 98) square units. However, we need to compute the actual side lengths using the perpendicular distances.
Step 6: Calculating the Side Lengths
The perpendicular distance between (11) and the line (4x 7y - 5 0) is used to find the length of a side:
[a frac{left|4 cdot 11 7 cdot 1 - 5right|}{sqrt{4^2 7^2}} Rightarrow frac{41 7 - 5}{sqrt{16 49}} Rightarrow frac{43}{sqrt{65}} Rightarrow 43 / sqrt{65}]
The perpendicular distance between (-31) and the line (7x - 4y 25 0) is used to find the length of another side:
[b frac{left|7 cdot (-3) - 4 cdot 1 - 25right|}{sqrt{7^2 4^2}} Rightarrow frac{-21 - 4 - 25}{sqrt{49 16}} Rightarrow frac{-50}{sqrt{65}} Rightarrow 50 / sqrt{65}]
Step 7: Final Calculation of the Area
The area of the rectangle is calculated as the product of the side lengths:
[text{Area} a times b frac{43}{sqrt{65}} times frac{50}{sqrt{65}} frac{2150}{65} 448 / 65]
Conclusion
We have successfully determined the sides and area of the rectangle using the given linear equations. The area of the rectangle is (boxed{frac{448}{65}}) square units.