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The (x, y) coordinates of points P and Q are

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The (x, y) coordinates of points P and Q are [#permalink] New post 19 Feb 2011, 15:26
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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?


a. 169 sqrt(3)/3
b. 84.5
c. 75 sqrt3
d. 169 sqrt(3)/4
e. 225 sqrt(3)/4

did not get it right. Can some explain this please.
[Reveal] Spoiler: OA

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Re: The (x, y) coordinates of points P and Q are [#permalink] New post 19 Feb 2011, 16:28
Bunuel, please could you explain the concept behind this ques .
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Re: The (x, y) coordinates of points P and Q are [#permalink] New post 19 Feb 2011, 18:24
Expert's post
ajit257 wrote:
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?
a. 169 sqrt(3)/3
b. 84.5
c. 75 sqrt3
d. 169 sqrt(3)/4
e. 225 sqrt(3)/4

did not get it right. Can some explain this please.


The formula to calculate the distance between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\).

So the distance between P and Q is \(d=\sqrt{(-2+7)^2+(9+3)^2}=13\).

So we know that the height in equilateral triangle XYZ equals to 13. Now, the height of an equilateral triangle equals to \(h=a\frac{\sqrt{3}}{2}\), where \(a\) is the side (as the height of the equilateral triangle divides it into two 30-60-90 right triangles with the ratio of the sides \(1:\sqrt{3}:2\) then height becomes the leg opposite 60 degrees angle and the hypotenuse, which is the side of an equilateral triangle can be found from the ratio: \(h=a\frac{\sqrt{3}}{2}\)) --> \(h=a\frac{\sqrt{3}}{2}=13\) --> \(a=\frac{26}{\sqrt{3}}\) --> \(area_{equilateral}=a^2*\frac{\sqrt{3}}{4}=\frac{169\sqrt{3}}{3}\).

Answer: A.

Check this: math-triangles-87197.html
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Re: The (x, y) coordinates of points P and Q are [#permalink] New post 19 Feb 2011, 19:14
thanks a ton Bunuel
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Re: The (x, y) coordinates of points P and Q are   [#permalink] 19 Feb 2011, 19:14
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