Inside a triangle formed by the line 3x+4y=1 and the positive coordinate axis, inscribe a square with three of the vertices on the coordinate axes and one vertex on the line.
1) Find the length of a side of the square.
In the triangular region described above inscribe a rectangle that has one vertex in the origin of the coordinate system and another vertex - on the line. Find the dimensions of this rectangle, if it is twice as wide as it is tall.
you want the width equal to the height. So, y=x and
x = (1-3x)/4
4x = 1-3x
x = 1/7
For the rectangle, you want x = 2y
so solve a similar equation.
To find the length of a side of the square inscribed in the triangle, we need to find the intersection points of the line 3x+4y=1 and the positive coordinate axes.
First, let's find the intersection point of the line and the x-axis:
Setting y = 0 in the equation 3x + 4y = 1:
3x + 4(0) = 1
3x = 1
x = 1/3
So, one vertex of the square is at the point (1/3, 0).
Next, let's find the intersection point of the line and the y-axis:
Setting x = 0 in the equation 3x + 4y = 1:
3(0) + 4y = 1
4y = 1
y = 1/4
Another vertex of the square is at the point (0, 1/4).
Since the square has three vertices on the coordinate axes, the last vertex can be obtained by reflecting one of the aforementioned vertices across the line 3x+4y=1.
Let's reflect the vertex (1/3, 0) across the line 3x+4y=1:
The perpendicular distance from the line to the vertex (1/3, 0) is given by the formula:
distance = |3(1/3) + 4(0) - 1| / sqrt(3^2 + 4^2)
distance = |1-1| / 5
distance = 0 / 5
distance = 0
Since the distance is 0, the line intersects the vertex and its reflection lies on the line as well. Therefore, the reflected vertex is also (1/3, 0).
Now we have three vertices of the square: (1/3, 0), (0, 1/4), and (1/3, 0).
Using the distance formula, we can find the length of the side of the square:
side length = sqrt((1/3 - 0)^2 + (0 - 1/4)^2)
side length = sqrt(1/9 + 1/16)
side length = sqrt(16/144 + 9/144)
side length = sqrt(25/144)
side length = 5/12
So, the length of a side of the square inscribed in the triangle is 5/12.
To find the dimensions of the rectangle inscribed in the triangle, we'll need to use the fact that it is twice as wide as it is tall.
Let's set the width of the rectangle as x and the height as y.
Since one vertex of the rectangle is at the origin (0,0), the other vertex lies on the line 3x + 4y = 1.
Substituting x=0 and y=0 into the equation of the line, we get:
3(0) + 4(0) = 1
0 = 1
This is not possible, so the assumption that the rectangle has one vertex at the origin is incorrect.
Therefore, there is no rectangle that satisfies both conditions - having a vertex at the origin and another vertex on the line.