In the figure above, points P and Q lie on the circle with center O. What is the value of s?
A) 1/2
B) 1
C) √2
D) √3
E) √2/2
The slope of a line is defined as rise/run, or the (change in y)/(the change in x).
The slopes of lines that are perpendicular to each other are the negative reciprocals of each other, meaning (the slope of one of the lines) = -1/(the slope of the other line).
In this case we have two line segments that are parts of lines that are perpendicular to each other and meet at O.
For the line passing through P we see that the change in y from O to point P is 1. The change in x is -√3.
So the slope of that line is 1/-√3 or -1/√3.
This means that the slope of the line passing through O and Q is -1/(-1/√3) or √3, and that for that line, every time x increases by 1, y increases by √3.
One the left side of the diagram we could draw a right triangle using line segment OP as the hypotenuse, to create a triangle with vertical side and height 1 and base √3.
We could also draw a right triangle using the segment OQ as the hypotenuse.
Since the distances from the origin to any point on the circle are the same, the distance from the origin to point Q is the same as the distance from the origin to point P. So a right triangle whose hypotenuse is the line segment OQ will have a hypotenuse the same length as segment OP.
Because the segment OP has slope -1/√3 and segment OQ has slope √3, we can figure out that the right triangle whose hypotenuse is OQ has the same dimensions as the triangle whose hypotenuse is segment OP, except that the height of the triangle on the left is 1 and the base is √3 and the height of the triangle on the right is √3 and the base is 1.
So s is
1 and the correct answer is
B.
