One side of the rectangle lies along the line 4x+7y+5=0 , two of its vertices are (-3,1)and (1,1). find
(i) the equations of the other three sides.
(ii) the angle between the two diagonals of the rectangle .
(iii) the area of the rectangle.
(i) To find the equations of the other three sides of the rectangle, we can first find the equations of the diagonals of the rectangle.
1. Find the midpoint of the line joining the given vertices (-3,1) and (1,1).
Midpoint formula: ((x1 + x2)/2, (y1 + y2)/2)
Midpoint = ((-3 + 1)/2, (1 + 1)/2) = (-1,1)
The diagonal of the rectangle passes through this midpoint, so we can use this point along with one of the given vertices to find the equation of this diagonal.
2. Find the equation of the diagonal passing through (-1,1) and (-3,1).
Slope formula: m = (y2 - y1) / (x2 - x1)
Slope = (1 - 1) / (-1 - (-3)) = 0
Since the slope is zero, the diagonal is a horizontal line passing through the midpoint.
The equation of the diagonal is y = 1.
3. Find the equation of the other diagonal using the same approach.
Diagonal passing through (-1,1) and (1,1):
Slope = (1 - 1) / (1 - (-1)) = 0
Equation: y = 1
4. To find the equations of the other two sides, we need to find the equations of two lines that are perpendicular to the diagonals and pass through the corner points (-3,1) and (1,1).
The lines will have slopes that are negative reciprocals of the slopes of the diagonals.
Diagonal 1 slope: 0, so the perpendicular line will have an undefined slope (vertical line).
Equation of the line passing through (-3,1): x = -3
Diagonal 2 slope: 0, so the perpendicular line will have an undefined slope (vertical line).
Equation of the line passing through (1,1): x = 1
The equations of the other three sides are:
Side 1: y = 1 (diagonal 1)
Side 2: x = -3 (perpendicular line at (-3,1))
Side 3: x = 1 (perpendicular line at (1,1))
(ii) To find the angle between the two diagonals, we can use the fact that the diagonals of a rectangle are perpendicular to each other.
The equation of the first diagonal is y = 1, and the equation of the second diagonal is also y = 1.
Since both diagonals are horizontal lines, they are parallel, and their slopes are equal.
The angle between two parallel lines is 0 degrees.
(iii) To find the area of the rectangle, we can use the distance formula to find the length and width of the rectangle.
The length is the distance between (-3,1) and (1,1):
Length = sqrt((1 - (-3))^2 + (1 - 1)^2)
= sqrt(16 + 0)
= sqrt(16)
= 4
The width is the distance between (-3,1) and (-3,1):
Width = sqrt((-3 - (-3))^2 + (1 - 1)^2)
= sqrt(0 + 0)
= sqrt(0)
= 0
Since the width is 0, the rectangle is actually a line segment. The area of a line segment is 0.
Therefore, the area of the rectangle is 0.
In summary:
(i) The equations of the other three sides are:
Side 1: y = 1
Side 2: x = -3
Side 3: x = 1
(ii) The angle between the two diagonals is 0 degrees.
(iii) The area of the rectangle is 0.
To find the equations of the other three sides of the rectangle, we need to first find the remaining two vertices of the rectangle.
(i) Finding the remaining vertices:
Let's find the intersection point of the given line and the line passing through (-3,1) and (1,1).
The line passing through (-3,1) and (1,1) has a slope of (1-1) / (1 - (-3)) = 0.
Since this line is parallel to the x-axis, the equation of the line passing through (-3,1) and (1,1) is y = 1.
Substituting y = 1 into the equation of the given line, we have:
4x + 7(1) + 5 = 0
4x + 12 = 0
4x = -12
x = -3
So, the intersection point is (-3,1).
Now, let's find the second vertex using the fact that diagonals of a rectangle are equal in length and bisect each other. The midpoint of (-3,1) and (1,1) is (-1,1).
Since the diagonals of a rectangle bisect each other and (-3,1) is one of the vertices, the other vertex must be the reflection of (-3,1) with respect to the line passing through the midpoint.
The line passing through the midpoint (-1,1) with slope 0 (since it is parallel to the x-axis) is y = 1.
The reflection of (-3,1) with respect to y = 1 is (3,1).
So, the remaining two vertices of the rectangle are (-1,1) and (3,1).
Now, let's find the equations of the other three sides of the rectangle:
Side 1: The side passing through (-3,1) and (-1,1) has a slope of (1-1) / (-1 - (-3)) = 0
Since this line is parallel to the x-axis, the equation of side 1 is y = 1.
Side 2: The side passing through (-1,1) and (3,1) also has a slope of (1-1) / (3 - (-1)) = 0.
So, the equation of side 2 is also y = 1.
Side 3: The side passing through (3,1) and (-3,1) is parallel to the y-axis and has the equation x = 3.
Therefore, the equations of the other three sides of the rectangle are:
Side 1: y = 1
Side 2: y = 1
Side 3: x = 3
(ii) Finding the angle between the diagonals:
To find the angle between the diagonals, we can use the fact that opposite sides of a rectangle are parallel and equal in length.
The diagonals of the rectangle are the line segments connecting the opposite vertices of the rectangle. In this case, the diagonals are the line segments connecting (-3,1) with (3,1), and (-1,1) with (1,1).
The two diagonals are horizontal lines since they have the same y-coordinates for both endpoints.
The angle between two horizontal lines is 0 degrees.
So, the angle between the two diagonals of the rectangle is 0 degrees.
(iii) Finding the area of the rectangle:
The area of a rectangle can be found by multiplying the length and width of the rectangle. In this case, the length of the rectangle is the distance between (-3,1) and (-1,1), and the width is the distance between (-1,1) and (3,1).
Using the distance formula, the length is:
√[(-1 - (-3))² + (1 - 1)²] = √[4 + 0] = √4 = 2
The width is:
√[(3 - (-1))² + (1 - 1)²] = √[16 + 0] = √16 = 4
Therefore, the area of the rectangle is 2 * 4 = 8 square units.
the given line has slope = -4/7
so, the perpendicular lines have slope 7/4
If we label the corners
A = (-3,1)
B
C = (1,1)
D
Then for side AB, (y-1)/(x+3) = 7/4
for side CD, (y-1)/(x-1) = 7/4
That will give you two lines; find where they intersect each other for B, and where CD intersects the given line for D.
Then find the lengths of the sides to get the diagonals and the area.
(We already know the diagonals have length 4 from AC.)
Yeah, it's some work, but this should get you started.