Geometry

10.7 Locus

not Locust!

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Goals

Know what Locus is. Find the locus given several

conditions.

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What is a LOCUS?

A set of points that satisfy some given condition.

Root is the Latin word for “Location”. Plural of Locus is Loci (low-sigh). Thinking of locus as a path may help.

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Locus Example 1

What is the locus of points on a plane equidistant from a given point?

A circle.

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Locus Example 1

What is the locus of points on a plane equidistant from a given point?

A circle.

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Locus Example 2

What is the locus of points equidistant from two given points?

How do we describe this line?

It’s the perpendicular bisector of the segment between the two points.

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Locus Example 3 – you try it.

What is the locus of points on a plane equidistant from line L?

L

How would you describe this?

Two lines parallel to L, one on each side.

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Locus Example 4 – you do it.

What is the locus of points equidistant from two perpendicular lines?

Description?

Two perpendicular lines that are the angle bisectors of the original lines.

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How to Determine a Locus

Locate a number of points which satisfy the given condition(s).

Draw a line (or curve) through these points.

Describe accurately the figure you have drawn.

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Problems to solve.

Do not talk about it. Do not ask questions. Think! Try to visualize in your “minds-eye”. Doodle or sketch if it helps. Write down a precise, accurate

description

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Problem 1

Find the locus of the midpoints of all chords that can be drawn from a fixed point in the circumference of a circle.

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Problem 1 Solution

The locus is a circle that is internally tangent to the given circle and with half of the diameter.

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Problem 2

A point moves so that it is always outside a square 3 ft. on a side and so that it is always 2 ft. from the nearest point of the square. Find the area enclosed by the locus of this moving point.

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Problem 2 Solution

3

32

2

2

Area

Square:

3 3 = 9

Rectangles:

4(2 3) = 24

Circle:

22 = 4

Total = 9 + 24 + 4

45.57

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Problem 3

Find the locus of the center of a circle which is externally tangent to each of two given non-intersecting and congruent circles.

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Problem 3 Solution

The perpendicular bisector of the segment between the centers of the two given circles.

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You cannot do locus problems without THINKING.