Area of an equilateral triangle- PS question

[vani_13in]

Can someone please help me solve this questions? I have the answer but dont know how it is arrived.

The (x, y) coordinates of points P and Q are (-2, 9) and (-7, -3), respectively. The height of equilateral triangle XYZ is the same as the length of line segment PQ. What is the area of triangle XYZ?


-V

[truplayer256]

Calculate the distance between P and Q to find the height of equilateral triangle XYZ.

Height of equilateral triangle XYZ =sqrt((-7+2)^(2)+(-3-9)^(2))=sqrt(25+144)=sqrt(169)=13.

The height of equilateral triangle XYZ splits the triangle into two 30-60-90 right triangles.

We know that a 30-60-90 triangle's sides are in the ratio s:s*sqrt(3):2s where s is one of the equilateral triangle's sides.

So, if the height of triangle XYZ is 13, then:

s*sqrt(3)=13

side of equilateral triangle=13sqrt(3)/3

Area of equilateral triangle XYZ=s^2*sqrt(3)/4=13^2*3*sqrt(3)/36=14.083*sqrt(3).

[vani_13in]

The answer is 169 sqrt 3 / 3.

Here is the explanation which I have not understood ( the part which says that short leg = 13 / sqrt 3)

We are told that this length (13) is equal to the height of the equilateral triangle XYZ. An equilateral triangle can be cut into two 30-60-90 triangles, where the height of the equilateral triangle is equal to the long leg of each 30-60-90 triangle. We know that the height of XYZ is 13 so the long leg of each 30-60-90 triangle is equal to 13. Using the ratio of the sides of a 30-60-90 triangle (1:sqrt 3: 2), we can determine that the length of the short leg of each 30-60-90 triangle is equal to 13/sqrt 3. The short leg of each 30-60-90 triangle is equal to half of the base of equilateral triangle XYZ. Thus the base of XYZ = 2(13/sqrt 3) = 26/sqrt 3.

-V