An equilateral triangle ⌂ABC
has two of its vertices below the x-axis,
has the third vertex C above the x-axis, and
contains the points A=(0,0)and B=(1,0) on its sides.
How long is the path traced out by all possible points C, to two decimal places?
so, two of its vertices are
(-h,-k) and (1+h,-k)
Thus, the length of the base is (2h+1)
That means the altitude is (1/2 + h)√3, making its coordinates (3/2, k+(1/2 + h)√3)
See whether you can work out the relationship between h and k, and thus the equation for the path followed by C.
Nah - that's just too complicated.
Take a look at an equilateral triangle, and slice off the top of it. (That's the piece of the x-axis between the sides of the triangle.) That small cap of the big triangle is also an equilateral triangle, and always has the same base length.
So, point C stays the same, no matter how big the triangle!
OK - I may have misread it again, assuming that the base of the triangle is parallel to the x-axis.
This is much trickier than it looks, unless the size of the triangle is somehow determined.
To calculate the length of the path traced out by all possible points C, we need to determine the range of possible positions for C and then calculate the sum of the distances between all these positions.
Given that two vertices of the equilateral triangle, A and B, are (0, 0) and (1, 0) respectively, we can deduce that the third vertex, C, lies somewhere on the line y = √3x ± d, where d is the perpendicular distance between the line and the x-axis.
Since C can be above the x-axis, we consider the equation y = √3x + d. To find the distance d, we substitute the coordinates of vertex B into the equation:
0 = √3(1) + d
d = -√3
Therefore, the equation of the line above the x-axis is y = √3x - √3.
Next, we consider the equation y = √3x + d for a line below the x-axis. Here, d should be positive. To find d, we substitute the coordinates of vertex A into the equation:
0 = √3(0) + d
d = 0
Therefore, the equation of the line below the x-axis is y = √3x.
To calculate the length of the path traced out by all possible points C, we need to find the range of x-values where the two lines intersect. Setting the equations equal to each other, we have:
√3x - √3 = √3x
√3 - √3 = √3x - √3x
0 = 0
As the equation is always true, the lines are coincident, suggesting that there is no range of x-values where the lines intersect. Therefore, there is no possible point C that satisfies the given conditions, and thus the path traced out by all possible points C has a length of 0.