2 problem solving difficulties

Q: 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)Root 2
d)Root 3
e)(Root 2)/2

The sides of a 30-60-90 triangle are proportioned x : x√3 : 2x.
Whenever you see √3 in a geometry question, look for -- or DRAW -- a 30-60-90 triangle.
Given (-√3, 1), y=1 and |x|=√3 represent the legs of a 30-60-90 triangle in which the sides are 1:√3:2.
Thus, we get the following drawing:



The correct answer is B.

A boat traveled upstream a distance of 90 miles at an average speed of
(v-3) miles per hour and then traveled the same distance downstream at an
average speed of (v+3) miles per hour. If the trip upstream took half an
hour longer than the trip downstream, how many hours did it take the boat
to travel downstream?

a) 2.5
b) 2.4
c) 2.3
d) 2.2
e) 2.1

We can PLUG IN THE ANSWERS, which represent the number of hours that the boat took to travel downstream.
When the correct answer choice is plugged in, the time upstream will be 1/2 hour longer than the time downstream.
The most likely answer choice is A -- the only option that divides evenly into 90.

Answer choice A: 2.5 hours to travel downstream.
Rate downstream = d/t = 90/(2.5) = 36 miles per hour.
Thus, v+3 = 36, implying that v=33.
Rate upstream = v-3 = 33-3 = 30 miles per hour.
Time upstream = d/r = 90/30 = 3 hours.
Time upstream - time downstream = 3-2.5 = .5.
Success!

The correct answer is A.

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

Another logical and very short method to solve this question is as follows

Slope of line OP = -(1/√3)
Product of slopes of perpendicular lines = -1
therefore slope of line OQ = √3/1

Since OP and OQ are both radii and the co ordinates of P are representing the slope without canceling any common factor between numerator and denominator therefore the slope of OQ Also will also represent the co-ordinates of the point Q [Furthermore this can also be verified by taking the distance between O and Q which becomes equal to radius=2 of the circle]

Slope of OQ = t/s = √3/1

therefore (s,t) = (1,√3)
therefore,
s = 1

Answer Option B