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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