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05 Jul 2021, 03:15
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E
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Difficulty:
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Question Stats:
67% (02:08) correct
33%
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wrong
based on 417
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In the xy-plane, an equilateral triangle has vertices at (0, 0) and (9, 0). What could be the coordinates of the third vertex?
(A) (0, 4.5)
(B) (4.5, 4.5)
(C) \((\frac{9\sqrt{3}}{2}, \ \frac{9\sqrt{3}}{2})\)
(D) \((4.5, \ 9\sqrt{3})\)
(E) \((4.5, \ \frac{9\sqrt{3}}{2})\)
(A) (0, 4.5)
(B) (4.5, 4.5)
(C) \((\frac{9\sqrt{3}}{2}, \ \frac{9\sqrt{3}}{2})\)
(D) \((4.5, \ 9\sqrt{3})\)
(E) \((4.5, \ \frac{9\sqrt{3}}{2})\)
05 Jul 2021, 03:29
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Given: In the xy-plane, an equilateral triangle has vertices at (0, 0) and (9, 0).
Asked: What could be the coordinates of the third vertex?
Side of the equilateral triangle = 9
Height of the equilateral triangle = \(\sqrt{81 - 81/4} = 9\sqrt{3}/2\)
Coordinates of the third vertex = \((4.5, 9\sqrt{3}/2)\)
IMO E
Asked: What could be the coordinates of the third vertex?
Side of the equilateral triangle = 9
Height of the equilateral triangle = \(\sqrt{81 - 81/4} = 9\sqrt{3}/2\)
Coordinates of the third vertex = \((4.5, 9\sqrt{3}/2)\)
IMO E
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Here is a much simpler explanation and solution for this seemingly problem.
I prefer to use distance formula instead of height or triangle formulas. Simply, because it is easy to remember and apply too.
Just remember these 2 things:
1. All sides of Equilateral Triangle are equal
2. Distance formula: Under root[ (x-x1)^2 + (y-y1)^2 ]
Let's find the length of one side with distance formula:
(9)^2 + 0 = under root (81) = 9 {Side length}
Now here's something simple:
Apply distance formula to the coordinates of the point to be found (Keep coordinates: x,y)
Here is what you will get:
x^2 + y^2 = (9-x)^2 + y^2 {Distance formula from x,y to 0,0 and x,y to 9,0}
Solving this, cancel out y^2
Finally, you will get a value of x as 9/2 i.e. 4.5
This eliminates 2 options and now you can simply find y by substituting x as 9/2.
Apply this from x,y to 0,0 with distance as 9.
Solving it will come down to:
81/4 + y^2 = 81
Solve it y^2 = 243/4
Remove the square and y = 9.under root(3)/4
Hence the option (E)
I prefer to use distance formula instead of height or triangle formulas. Simply, because it is easy to remember and apply too.
Just remember these 2 things:
1. All sides of Equilateral Triangle are equal
2. Distance formula: Under root[ (x-x1)^2 + (y-y1)^2 ]
Let's find the length of one side with distance formula:
(9)^2 + 0 = under root (81) = 9 {Side length}
Now here's something simple:
Apply distance formula to the coordinates of the point to be found (Keep coordinates: x,y)
Here is what you will get:
x^2 + y^2 = (9-x)^2 + y^2 {Distance formula from x,y to 0,0 and x,y to 9,0}
Solving this, cancel out y^2
Finally, you will get a value of x as 9/2 i.e. 4.5
This eliminates 2 options and now you can simply find y by substituting x as 9/2.
Apply this from x,y to 0,0 with distance as 9.
Solving it will come down to:
81/4 + y^2 = 81
Solve it y^2 = 243/4
Remove the square and y = 9.under root(3)/4
Hence the option (E)
