An equilateral triangle has sides 12 cm.It is divided into three triangles of?

Question

An equilateral triangle has sides 12 cm.It is divided into three triangles of equal area by two line segments passing through one of the vertices. Determine the lengths of these lines.

Answer 1

in order to have 3 triangles of equal area, the 2 line segments must divide the base into 3 equal parts.

If you draw that out, you can see that the middle traingle is an isosceles triangle. To figure out the length of the two line segments, you can utilize pythagoras theorem by drawing a perpendicular line from the tip of the middle triangle down to the base (i.e. the height of the triangle) ==> sqrt (12^2 - 6^2) = 6sqrt(3).

Again, using pythagoras theorem, the side of the isosceles triangle in the middle ==> sqrt (2^2 + (6sqrt(3))^2) = sqrt (112) = 4 sqrt(7) <-- that's your answer! :)

Answer 2

The lines divide the base into 3 equal parts since all 3 triangles have the same height.
You have (2) triangles with sides 12 & 4 with a 60° angle in between them.
from the law of cosines, the 3rd sid is
x^2=12^2+4^2-2*4*12*cos 60°
x^2=160-96 cos 60°
x^2=112
x=4√7

Answer 3

area being 1/2 * base * height, and as the lines are drawn from the top vertex to the base of the equilateral triangle such that there are 3 triangles of base 4 cm each and the height the same as the one for the main triangle, sqrt ( (12 ^ 2) - (6 ^ 2)) = sqrt (108).
Now, this altitude from the top vertex to the center of the 12 cm base is a median of the central triangle, whose 4 cm base is divided into 2 cm each by the altitude and each of the other sides are the lines that have divided the main triangle into 3 equal areas.
Each of them will be sqrt ( 108 + (2 ^ 2) ) = sqrt (112).

Answer 4

Equilateral triangle: all internal angles must be 60 degrees.

Imagine we take the top vertice and extend two lines to the opposite side, equally splitting the 60 into three parts: 20 degrees each respectively. We'll take a look at one triangle: the one on the left

The outer edge of the triangle is 12cm. The top angle will be 20, the bottom left angle will be 60, leaving the internal angle to be 100.

Next, we use the law of sines:

x / (sin 60) = 12 / (sin 100)
x = 12 (sin 60) / sin (100)

Which is roughly x = 10.55 cm.

Answer 5

Call "A" the area of this 12 cm equilateral triangle.

area = 6 x 10.392 = 62.352
1/3 x 62.352 = 20.784, area of 3 small triangles.

The center triangle will have the base of 1.333 the other two will have the base of (12 - 1.333)/2 = 5.333

Answer 6

Editing

Answer 7

Try drawing it.

Answer 8

is it 8 cm.?

Answer 9

4cm