Equilateral triangle ABC is inscribed in circle O, whose radius is 4. Altitude BD is extended until it intersects the circle at E. What is the length of DE?
a) √3
b) 2√3
c) 2
d) 4√3
The altitude of an equilateral triangle is also a median.
The medians intersect 2/3 of the way from the vertex to the opposite side.
So, BO=4, and BD=6
BE is the diameter of the circle, so BE=8
so, what do you think?
To find the length of DE, we can use the properties of a circle and an equilateral triangle.
First, let's consider triangle BDE. Since triangle BDE is a right triangle (formed by the altitude BD and the extended BD line), we can use the Pythagorean theorem to solve for DE.
Let's assign a variable to the unknown length DE. Let's say DE = x.
Now, in triangle BDE, using the Pythagorean theorem, we have:
BD^2 + DE^2 = BE^2
Since triangle ABC is an equilateral triangle, all sides are equal. Hence, BD = BC. Since BC is the radius of circle O, which is given as 4, we have:
(4)^2 + x^2 = BE^2
16 + x^2 = BE^2
Next, observe triangle BEO. Since triangle BEO is also an equilateral triangle (since all sides are radii of the circle), we can determine BE.
Using properties of an equilateral triangle, we know that the length of an altitude of an equilateral triangle is given by:
altitude = (sqrt(3) / 2) * side length
Since the side length of triangle BEO is BE, we have:
BE = (sqrt(3) / 2) * BO
BE = (sqrt(3) / 2) * 4
BE = 2sqrt(3)
Now, substitute this value of BE into the equation we obtained earlier:
16 + x^2 = (2sqrt(3))^2
16 + x^2 = 12
x^2 = 12 - 16
x^2 = -4
Since x^2 is negative, this implies that there is no real solution for x. Therefore, the length of DE cannot be determined based on the given information.
The answer is: It is not possible to determine the length of DE based on the given information.