The base of a triangle is on the x-axis, one side lies along the line y=3x, and the third side passes through the point (1,1). What is the slope of the third side if the area of the triangle is to be a minimum.
Interesting problem. Check to make sure I get this right.
So, we have a triangle with base k, and vertex where the line through (1,1) and (0,k) intersects y=3x.
The line through (1,1) and (0,k) is
(y-1)/(x-1) = -1/(k-1)
y = (x-1)/(1-k) + 1
So, that line intersects y=3x when
3x = (x-1)/(1-k) + 1
x = k/(3k-2)
y = 3k/(3k-2)
The area a of the triangle is thus
a = 1/2 k * 3k/(3k-2) = 3k^2/(6k-4)
da/dk = (6k(6k-4) - 3k^2 * 6)/(6k-4)^2
= 6k(3k-4)/(6k-4)^2
da/dk = 0 when k=0 or k = 4/3
Thus, the vertex of the triangle is where the line through (1,1) and (4/3,0) intersects y=3x
(y-1)/(x-1) = 1/(-1/3)
y = -3x + 4
That line has slope -3
. . .
-3x + 4 = 3x
6x = 4
x = 2/3
y=2
So, the triangle has base k=4/3, height h=2, area a=4/3
To find the slope of the third side of the triangle, we need to determine the coordinates of the base of the triangle. Since the base lies on the x-axis, its y-coordinate is 0.
Now, let's find the x-coordinate of the base. Since the side of the triangle lies along the line y = 3x, the coordinates of the side can be written as (x, 3x).
Since the side also passes through the point (1,1), we can equate the x and y coordinates to find the x-coordinate of the base:
x = 1 (equating x-coordinates)
3x = 1 (equating y-coordinates)
Solving the second equation for x, we get:
x = 1/3
Therefore, the coordinates of the base of the triangle are (1/3, 0).
Now, to find the slope of the third side, we need to determine the coordinates of the third side. Since the third side passes through the point (1,1), let's assume its coordinates are (x, y).
We can use the formula for the slope of a line to find the slope (m) of the third side:
m = (y - 1)/(x - 1)
To find the minimum area of the triangle, we can use the formula for the area of a triangle:
Area = (1/2) * base * height
We know that the base of the triangle is the distance between the two vertices on the x-axis, which is given by:
base = |(1/3) - 1|
Simplifying, we have:
base = |2/3|
Since we want to minimize the area of the triangle, we need to minimize the height. The height is the vertical distance from the third side to the x-axis.
To find the height, we can use the equation of the line y = 3x. Substituting the x-coordinate of the third side, we get:
height = 3 * (x - 1)
Now, substituting the values of the base and the height into the formula for the area, we have:
Area = (1/2) * (2/3) * 3 * (x - 1)
Simplifying, we get:
Area = x - 1
To minimize Area, we need to minimize x. Since the x-coordinate of the third side lies on the x-axis, its value is 0.
Substituting x = 0 into the slope formula, we get:
m = (0 - 1)/(0 - 1)
m = -1/(-1)
m = 1
Therefore, the slope of the third side of the triangle, when the area is minimized, is 1.
To find the slope of the third side of the triangle that will minimize its area, we need to determine the equation of the third side first.
Let's consider the information given:
- The base of the triangle lies on the x-axis, meaning that it has a y-coordinate of 0.
- One side of the triangle lies along the line y = 3x.
- The third side passes through the point (1,1).
Since the base lies on the x-axis, it means that its two vertices have coordinates (x1, 0) and (x2, 0). Let's assume x1 and x2 are the x-coordinates of these two vertices.
We know that the equation of the line passing through two points (x1, y1) and (x2, y2) can be found using the slope-intercept form: y - y1 = m(x - x1), where m is the slope of the line.
Now, let's find the slope m of the line y = 3x, which is given as one side of the triangle. Comparing the equation y = 3x with the slope-intercept form y = mx + b, we can identify that the slope m is 3.
We also know that the third side of the triangle passes through the point (1, 1). Therefore, one of the vertices of the third side is (1, 1).
Now, let's substitute the values we have into the slope-intercept form equation:
y - 1 = m(x - 1)
Replacing m with the slope of the line, we get:
y - 1 = 3(x - 1)
Expanding and rearranging the equation, we can rewrite it:
y = 3x - 3 + 1
Simplifying further:
y = 3x - 2
The slope of the third side of the triangle is 3, as it appears as the coefficient of x in the equation y = 3x - 2.
Therefore, the slope of the third side of the triangle that minimizes its area is 3.