Question

An equilateral triangle has a side length of 6. What is the height of the triangle?

Answer 1

Since the height of an equilateral triangle in terms of its side s is s√3/2, the height of the triangle is 6√3/2 = 3√3 and so the area is (1/2)(6)(3√3) = 9√3. 

If we draw a horizontal line a height of h from the base of the triangle, the region is split into two regions: the lower region consisting of a trapezoid of height h and the upper region consisting of a triangle of height 3√3 - h. 

Since the upper triangle and the triangle itself are similar triangles, the base and height are proportional. If we let x denote the base of the length of the upper triangle, we have: 

(S. of small triangle)/(S. of big triangle) = (Ht. of small triangle)/(Ht. of big triangle) 
==> x/6 = (3√3 - h)/(3√3) 
==> x = (6√3 - 2h)/√3 

Thus, the area of the upper triangle is: 

A = (1/2)[(6√3 - 2h)/√3](3√3 - h) = [(6√3 - 2h)(3√3 - h)]/(2√3). 
(Made a dumb mistake about the height here for some reason) 

Since we require that the area of this triangle is to be half of the total area (9√3/2), we need to solve: 

[(6√3 - 2h)(3√3 - h)]/(2√3) = 9√3/2 
==> (6√3 - 2h)(3√3 - h) = 27 
==> 54 - 6h√3 - 6h√3 + 2h^2 = 27 
==> 2h^2 - 12h√3 + 27 = 0. 

Solving with the Quadratic Formula gives: 

h = (6√3 + 3√6)/2 ≈ 8.87 units and h = (6√3 - 3√6)/2 ≈ 1.52 units. 

Since h = (6√3 + 3√6)/2 would place the line outside of the triangle, we pick h = (6√3 - 3√6)/2. 

Therefore, the line should be ==> (6√3 - 3√6)/2 units from the base. 

I hope this helps! ^^ Brainliest Please?

Answer 2

Answer:

The height of the triangles is  3* sqrt(3)

Step-by-step explanation:

To find the height of an equilateral triangles we can use the rule of 30-60-90 Triangles where:

Height : a* sqrt(3)

Shorter side: a

Hypotenuse: 2a

Where a is the side of the base or shorter side of the triangle. If you draw the height of an equilateral triangle to its base you bisect one of the angles in two 30 degrees angles. So we arrive to two  30 - 60 -90 triangles. The  height divides the side into segments of length 3. So following the 30-6090 rule, the measures of this triangles are:

Shorter side: 3

Hypotenuse: 6

Height : 3* sqrt(3)

This height is equivalent to the one of the equilateral triangle so the height of the equi triangle is  3* sqrt(3)