Find the lengths of a triangle whose two sides lie on the coordinate axes and the other side passes through the point (1,1)
I have seen this question, where we have to find the other sides so that the perimeter is a minimum, but ...
The way it is worded...
there is an infinite number of possible lengths.
e.g if the side along the x-axis is 5, then the equation of the line passing through (1,1) has slope of (0-1)/(5-1) = -1/4
equation: y -1 = (-1/4)(x-1)
4y - 4 = -x + 1
x + 4y = 5
y-intercept: let x=0 ----> y = 5/4
so the length along the y axis is 5/4
the length along the x-axis is 5
the slanted line has length √(5^2 + (5/4)^2)
= √(425/16)
= 5√17/4
there would be an infinite number of such calculations.
To find the lengths of a triangle whose two sides lie on the coordinate axes and the other side passes through the point (1, 1), we need to determine the coordinates of the other two vertices of the triangle.
Let's consider the side of the triangle that lies on the x-axis. Since one of the vertices is on the origin (0, 0) and the other side passes through the point (1, 1), the x-coordinate of the third vertex must be 0 to lie on the x-axis.
Now, let's consider the side of the triangle that lies on the y-axis. Since one of the vertices is on the origin (0, 0) and the other side passes through the point (1, 1), the y-coordinate of the third vertex must be 0 to lie on the y-axis.
Therefore, the coordinates of the three vertices of the triangle are (0, 0), (1, 1), and (0, 0).
To find the lengths of the sides of the triangle, we can use the distance formula which is based on the Pythagorean theorem:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Let's calculate the lengths of the sides of the triangle:
1. Length of the side on the x-axis:
Distance = sqrt((1 - 0)^2 + (1 - 0)^2) = sqrt(2)
2. Length of the side on the y-axis:
Distance = sqrt((0 - 0)^2 + (0 - 1)^2) = sqrt(1) = 1
3. Length of the side connecting (0, 0) to (1, 1):
Distance = sqrt((1 - 0)^2 + (1 - 0)^2) = sqrt(2)
Therefore, the lengths of the sides of the triangle are sqrt(2), 1, and sqrt(2).