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“Birds-Eye” View

Most geometry problems on the test require straightforward calculations. However, some problems measure your insight into the basic rules of geometry. For this type of problem, you should step back and take a “birds-eye” view of the problem. The following example will illustrate.
In the figure, O is both the center of the circle with radius 2 and a vertex of the square OPRS. What is the length of diagonal PS?
A.   1/2
C.   4
D.   2

The diagonals of a square are equal.
Hence, line segment OR (not shown) is equal to SP.
Now, OR is a radius of the circle and therefore OR = 2.
Hence, SP = 2 as well, and the answer is (D).


Tips: When Drawing a Geometric Figure or Checking a Given One, Be Sure to Include Drawings of Extreme Cases As Well As Ordinary Ones.


AC is a chord.
B is a point on the circle.
Column A Column B
x 45
Although in the drawing AC looks to be a diameter, that cannot be assumed. All we know is that AC is a chord. Hence, numerous cases are possible, three of which are illustrated below:
Case I Case II
Case III

In Case I, x is greater than 45 degrees; in Case II, x equals 45 degrees; in Case III, x is less than 45 degrees. Hence, the answer is (D).


Three rays emanate from a common point and form three angles with measures pqr.
Column A Column B
180 measure of q + r

It is natural to make the drawing symmetric as follows:
In this case, p = q = r = 120˚, so q + r = 240˚.
Hence, Column B is larger. However, there are other drawings possible.
For example:
In this case, q + r = 180˚ and therefore the two columns are equal. This is a double case, and the answer is (D)—not-enough-information.

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