slope of line connecting origin and 1, -sqrt-3 = -1/sqrt-3
Slope of perpendicular line = sqrt-3
Eqn of perpendicular line is y = sqrt-3*x. (s,t) is a point on this. So,
t = sqrt-3*s
Distance between s,t and origin = s^2 + t^2 = 4*s^2
This should be the same distance between 1, -sqrt-3 i.e. 4
so, 4*s^2 = 4. s = 1
You can also use properties of 30-60-90 triangles to solve this.
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jayhawk2001
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Hi,
The way you solved is completely unclear, can you explain this using the 30-60-90 method or break it down a bit more?
Thanks
The way you solved is completely unclear, can you explain this using the 30-60-90 method or break it down a bit more?
Thanks
see the attached picture.
Extending a line from point P down to the origin, you can see a right triangle of sides sqrt(3) and 1. General trig (SOH CAH TOA or 1,2,sqrt3 rule) will tell you that this is a 30 60 90 triangle. And the 1,2,sqrt(3) rule tells you that the hypotenuse is twice the shortest side. So the hypotenuse (i.e. the radius) is 2.
The angle opposite the side measured 1 is 30 degrees and the angle opposite the side measured sqrt(3) is 60 degrees (This is a general rule for a 30-60-90 triangle).
Extending another line from point Q down to the origin we have another right triangle. The angle in that triangle that's closest to the origin can be determined: 180 - (30 +90) = 60 degrees. Two angles of that right triangle are known so we can determine that the third angle is 30 degrees.
We have another 30-60-90 triangle with hypotenuse= 2 (remember it's the radius). So the shortest side (side opposite angle 30 degrees) is 1 (S) and the side opposite angle 60 degrees is sqrt(3) (T).
S is just the length of the shortest side which we determined to be 1. Hence B
Extending a line from point P down to the origin, you can see a right triangle of sides sqrt(3) and 1. General trig (SOH CAH TOA or 1,2,sqrt3 rule) will tell you that this is a 30 60 90 triangle. And the 1,2,sqrt(3) rule tells you that the hypotenuse is twice the shortest side. So the hypotenuse (i.e. the radius) is 2.
The angle opposite the side measured 1 is 30 degrees and the angle opposite the side measured sqrt(3) is 60 degrees (This is a general rule for a 30-60-90 triangle).
Extending another line from point Q down to the origin we have another right triangle. The angle in that triangle that's closest to the origin can be determined: 180 - (30 +90) = 60 degrees. Two angles of that right triangle are known so we can determine that the third angle is 30 degrees.
We have another 30-60-90 triangle with hypotenuse= 2 (remember it's the radius). So the shortest side (side opposite angle 30 degrees) is 1 (S) and the side opposite angle 60 degrees is sqrt(3) (T).
S is just the length of the shortest side which we determined to be 1. Hence B
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Last edited by jrbrown2 on Wed Nov 21, 2007 1:12 pm, edited 2 times in total.
Sol by 30-60-90 property: rem 30 - 60 - 90 then sides are 1 - root3 - 2
Please chk the attached image!
draw a perpendicular from point P on the x axis, lets call is as M ( -root3, 0)
triangle opm is right angle triangle. OP is the radius of the circle.
by pythagorean OP is 2....got it??
now triangle opm is 30-60-90 as the sides are 1-root3 -2
draw a perpendicular from point Q on the x axis, lets call is as N
angle pom + angle poq + angle qon = 180
30 + 90 + angle qon = 180
angle qon = 60
again triangle qon is 30-60-90 as the sides are 1-root3 -2
So S = 1
Please chk the attached image!
draw a perpendicular from point P on the x axis, lets call is as M ( -root3, 0)
triangle opm is right angle triangle. OP is the radius of the circle.
by pythagorean OP is 2....got it??
now triangle opm is 30-60-90 as the sides are 1-root3 -2
draw a perpendicular from point Q on the x axis, lets call is as N
angle pom + angle poq + angle qon = 180
30 + 90 + angle qon = 180
angle qon = 60
again triangle qon is 30-60-90 as the sides are 1-root3 -2
So S = 1
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- sol.doc
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