April 18, 2023 Geometry 10.7 Locus 2
Goals
Know what Locus is. Find the locus given several
conditions.
April 18, 2023 Geometry 10.7 Locus 3
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.
April 18, 2023 Geometry 10.7 Locus 4
Locus Example 1
What is the locus of points on a plane equidistant from a given point?
A circle.
April 18, 2023 Geometry 10.7 Locus 5
Locus Example 1
What is the locus of points on a plane equidistant from a given point?
A circle.
April 18, 2023 Geometry 10.7 Locus 6
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.
April 18, 2023 Geometry 10.7 Locus 7
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.
April 18, 2023 Geometry 10.7 Locus 8
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.
April 18, 2023 Geometry 10.7 Locus 9
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.
April 18, 2023 Geometry 10.7 Locus 11
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
April 18, 2023 Geometry 10.7 Locus 12
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.
April 18, 2023 Geometry 10.7 Locus 13
Problem 1 Solution
The locus is a circle that is internally tangent to the given circle and with half of the diameter.
April 18, 2023 Geometry 10.7 Locus 14
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.
April 18, 2023 Geometry 10.7 Locus 15
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
April 18, 2023 Geometry 10.7 Locus 16
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.
April 18, 2023 Geometry 10.7 Locus 17
Problem 3 Solution
The perpendicular bisector of the segment between the centers of the two given circles.