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
Area of an equilateral triangle- PS question
This topic has expert replies
-
- Master | Next Rank: 500 Posts
- Posts: 392
- Joined: Thu Jan 15, 2009 12:52 pm
- Location: New Jersey
- Thanked: 76 times
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).
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).
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
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