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# Geometry with a Compass Practice Problems

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— McGraw-Hill Professional
Updated on Oct 3, 2011

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## Geometry with a Compass Practice Problems

#### PROBLEM 1

Define a point by drawing a dot. Then, with the compass, draw a small circle centered on the dot. Now construct a second circle, concentric with the first one, but having twice the radius.

#### SOLUTION 1

Fig. 5-5 . Illustration for Problem 1.

Figure 5-5 illustrates the procedure. In drawing A, the circle is constructed with the compass, centered at the initial point (called point P ). In drawing B, a line segment L is drawn using the straight edge, with one end at point P and passing through the circle at a point Q . The line segment extends outside the circle for a distance considerably greater than the circle's radius. In drawing C, a circle is constructed, centered at point Q and leaving the compass set for the same radius as it was when the original circle was drawn. This new circle intersects L at point P (the center of the original circle) and also at a new point R . Next, the non-marking tip of the compass is placed back at point P , and the compass is opened up so the pencil tip falls exactly on point R . Finally, as shown in drawing D, a new circle is drawn with its center at point P , with a radius equal to the length of line segment PR .

#### PROBLEM 2

Draw three points on a piece of paper, placed so they do not all lie along the same line. Label the points P, Q , and R . Construct Δ PQR connecting these three points. Draw a circle whose radius is equal to the length of side PQ , but that is centered at point R .

#### SOLUTION 2

The process is shown in Fig. 5-6. In drawing A, the three points are put down and labeled. In drawing B, the points are connected to form Δ PQR . Drawing C shows how the non-marking tip of the compass is placed at point Q , and the tip of the pencil is placed on point P . (You don't have to draw the arc, but it is included in this illustration for emphasis.) With the compass thus set so it defines the length of line segment PQ , the non-marking tip of the compass is placed on point R . Finally, as shown in drawing D the circle is constructed.

#### PROBLEM 3

Can the non-marking tip of the compass be placed at point P , and the pencil tip placed to draw an arc through point Q , in order to define the length of line segment PQ in Problem 2?

#### SOLUTION 3

Yes. This will work just as well.

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