**Problem 1 :**

Draw point C on a piece of paper. Draw and describe the locus of all points on the paper that are e inches from c.

**Problem 2 :**

Points A and B lie in a plane. What is the locus of points in the plane that are equidistant from points A and B and are a distance of AB from B ?

**Problem 3 : **

Point P is in the interior of ∠ABC. What is the locus of points in the interior of ∠ABC that are equidistant from both sides of ∠ABC and 2 inches form P ? How does the location of P within ∠ABC affect the locus ?

**Problem 4 :**

We have the following readings of an earthquake from three seismographs.

(i) At A(-5, 5), the epicenter is 4 miles away.

(ii) At B(-4, -3.5), the epicenter is 5 miles away.

(iii) At C(1, 1.5), the epicenter is 7 miles away.

Where is the epicenter ?

**Problem 1 :**

Draw point C on a piece of paper. Draw and describe the locus of all points on the paper that are e inches from c.

**Solution : **

**Step 1 : **

Draw point C. Locate several points e inches
from C.

**Step 2 : **

Recognize a pattern: the points lie on a circle.

**Step 3 :**

Draw the circle using compass as shown below.

The locus of points on the paper that are e inches from C is a circle with center C and a radius of 3 inches.

**Problem 2 :**

Points A and B lie in a plane. What is the locus of points in the plane that are equidistant from points A and B and are a distance of AB from B ?

**Solution : **

The locus of all points that are equidistant from A and B is the perpendicular bisector of AB as shown below.

The locus of all points that are a distance of AB from B is the circle with center B and radius AB as shown below.

These loci intersect at D and E. So D and E are the locus of points that satisfy both conditions.

**Problem 3 : **

Point P is in the interior of ∠ABC. What is the locus of points in the interior of ∠ABC that are equidistant from both sides of ∠ABC and 2 inches form P ? How does the location of P within ∠ABC affect the locus ?

**Solution : **

The locus of points equidistant from both sides of ∠ABC is the angle bisector. The locus of points 2 inches form P is a circle. The intersection of the angle bisector and the circle depends on the location of P. The locus can be 2 points, 1 point, or 0 points.

**The locus is two points :**

**The locus is one point :**

**The locus is zero points :**

**Problem 4 :**

We have the following readings of an earthquake from three seismographs.

(i) At A(-5, 5), the epicenter is 4 miles away.

(ii) At B(-4, -3.5), the epicenter is 5 miles away.

(iii) At C(1, 1.5), the epicenter is 7 miles away.

Where is the epicenter ?

**Solution : **

Each seismograph gives us a locus that is a circle.

So, we have

Circle A has center (-5, 5) and radius 4

Circle B has center (-4, -3.5) and radius 5

Circle C has center (1, 1.5) and radius 7

Draw the three circles in a coordinate plane. The point of intersection of the three circles is the epicenter.

In the diagram above, three circles intersect at (-6, 1).

The epicenter is at about (-6, 1).

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