Question

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?

Answer 1

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.

Answer 2

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!

Answer 3

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.