# Loci in Geometry

High School Math based on the topics required for the Regents Exam conducted by NYSED. The basic locus theorem and and compound loci.

Lesson on the laws of locus and compound loci.

These lessons are aligned with the NYS Regents Geometry Curriculum.

Related Topics: More Geometry Lessons

## Examples

10) A treasure map shows a treasure hidden in a park near a tree and a stature. The map indicates that the tree and the statute are 10 ft apart. The treasure is buried 7 ft from the base of the tree and also 5 ft from the base of the statute. How many places are possible locations for the treasure to be buried? Draw a diagram of the treasure map and indicate with an X each possible location of the treasure.

11) The distance between parallel lines l and m is 12 units. Point A is on line l. How many points are equidistant from line l and m and 8 units from point A?

12) Mari's backyard has two trees that are 40 feet apart. She wants to place lampposts so that the posts are 30 ft from both of the trees. Draw a sketch to show where the lampposts could be placed in relation to the trees. How many locations for the lampposts are possible?

Loci 1

Loci 1 - describing and drawing the locus of all points following certain rules.

1) Locus of all points exactly 5 m from point P.

2) Locus of all points less than 5 m from point P.

3) The locus of all points that are within 3 m from the line AB.

4) The locus of all points inside the rectangle which are closer to A than to C.

5) The locus of all points inside the triangle
that are closer to the line AB than to the line AC.

Loci 2

More on locus of points following rules, this time when more than one rule is being followed.

1) A lawn is to be planted inside a garden, at least 4 m from the side of the house and at least 3 m fro the tree (T). Shade the lawn.

2) A rose bed is to be planted
within 5 m of A and and within 6 m of C. Shade the rose bed.

3) Caroline wants to plant a lawn
which is closer to side AB than AD, and within 6 m of C. Shade her lawn.

Locus Theorems:

**Locus Theorem 1**: The locus of points at a fixed distance, d, from point P is a circle with the given point P as its center and d as its radius.

**Locus Theorem 2**: The locus of points at a fixed distance, d, from a line, l, is a pair of parallel lines d distance from l and on either side of l.

**Locus Theorem 3:** The locus of points equidistant from two points, P and Q, is the perpendicular bisector of the line segment determined by the two points.

**Locus Theorem 4: ** The locus of points equidistant from two parallel lines, l1 and l2, is a line parallel to both l1 and l2 and midway between them.

**Locus Theorem 5:** The locus of points equidistant from two intersecting lines, l1 and l2, is a pair of bisectors that bisect the angles formed by l1 and l2.

You can use the Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.