On the portion of the straight line x+y-7=0 which is intercepted between the axes a square is constructed on the side og the line away from the origin then find the equqtions of the diagonals?
To find the equations of the diagonals of the square constructed on the side of the line x + y - 7 = 0 away from the origin, we need to break down the problem into steps:
Step 1: Find the coordinates of the vertices of the square
First, let's find the intercepts of the given line with the x and y axes.
To find the x-intercept, we set y = 0 and solve for x:
x + 0 - 7 = 0
x = 7
So, the x-intercept is (7, 0).
To find the y-intercept, we set x = 0 and solve for y:
0 + y - 7 = 0
y = 7
So, the y-intercept is (0, 7).
Now, we have two vertices of the square: the x-intercept (7, 0) and the y-intercept (0, 7).
Step 2: Find the remaining two vertices of the square
Since the square is constructed on the side of the line away from the origin, we need to determine the remaining two vertices. These vertices will form a line perpendicular to the given line x + y - 7 = 0 and pass through the x-intercept (7, 0) and y-intercept (0, 7).
The slope of the given line is -1 (since the coefficient of x and y are both 1). The slope of a line perpendicular to this line will be 1 (since it is the negative reciprocal of -1).
Using the point-slope form of a line, we can find the equation of the line passing through (7, 0) with a slope of 1:
y - 0 = 1(x - 7)
y = x - 7
Now, we need to find the point where this line intersects the x-axis by setting y = 0:
0 = x - 7
x = 7
So, the third vertex of the square is (7, 0).
By symmetry, the fourth vertex will be the reflection of the third vertex over the line x + y - 7 = 0. We can find the reflection by swapping the x and y coordinates of the third vertex:
The fourth vertex of the square is (0, 7).
Step 3: Find the equations of the diagonals
We have the coordinates of two opposite vertices of the square: (7, 0) and (0, 7).
To find the equation of the diagonal passing through these two points, we use the point-slope form of a line:
1. For the diagonal passing through (7, 0):
m = (y2 - y1) / (x2 - x1)
= (7 - 0) / (0 - 7)
= -1
Using the point-slope form, we can write the equation as:
y - 7 = -1(x - 0)
y - 7 = -x
Simplifying the equation, we get:
x + y - 7 = 0
2. For the diagonal passing through (0, 7):
m = (y2 - y1) / (x2 - x1)
= (0 - 7) / (7 - 0)
= -1
Using the point-slope form, we can write the equation as:
y - 0 = -1(x - 7)
y = -x + 7
Simplifying the equation, we get:
x + y - 7 = 0
Therefore, the equations of both diagonals are x + y - 7 = 0.