Theorems on Locus of a Point which is Equidistant from Two Fixed Points

The locus of a point which is equidistant from two fixed points is the perpendicular bisector of the line segment joining the two fixed points.

Given,

Let X and Y be two given fixed points. PQ is the path traced out by the moving point P such that each point on it is equidistant from X and Y. Therefore, PX = PY.



To prove: PQ is the perpendicular bisector of the line segment XY.

Construction: Join X to Y. Let PQ cut XY at O.

Theorem on Locus

Proof:

From △PXO and △PYO,

PX and PY (Given)

XO = YO (Since, every point of PQ is equidistant from X and Y, and O is a point on PQ.)

PO = PO (Common side.)

Therefore, by the SSS criterion of congruency△PXO ≅ △PYO.

Now ∠POX = ∠POY (since, corresponding parts of congruent triangles are congruent.)

Again ∠POX + ∠POY = 180° (Since, XOY is a straight line.

Therefore, ∠POX = ∠POY = \(\frac{180°}{2}\) = 90°

Also, PQ bisects XY (Since, XO = YO)

Therefore, PQ ⊥ XY and PQ bisects XY, i.e., PQ is the perpendicular bisector of XY (Proved)

 Loci












10th Grade Math

From Theorems on Locus of a Point which is Equidistant from Two Fixed Points to HOME




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



Share this page: What’s this?

Recent Articles

  1. Tangrams Math | Traditional Chinese Geometrical Puzzle | Triangles

    Apr 17, 24 01:53 PM

    Tangrams
    Tangram is a traditional Chinese geometrical puzzle with 7 pieces (1 parallelogram, 1 square and 5 triangles) that can be arranged to match any particular design. In the given figure, it consists of o…

    Read More

  2. Time Duration |How to Calculate the Time Duration (in Hours & Minutes)

    Apr 17, 24 01:32 PM

    Duration of Time
    We will learn how to calculate the time duration in minutes and in hours. Time Duration (in minutes) Ron and Clara play badminton every evening. Yesterday, their game started at 5 : 15 p.m.

    Read More

  3. Worksheet on Third Grade Geometrical Shapes | Questions on Geometry

    Apr 16, 24 02:00 AM

    Worksheet on Geometrical Shapes
    Practice the math worksheet on third grade geometrical shapes. The questions will help the students to get prepared for the third grade geometry test. 1. Name the types of surfaces that you know. 2. W…

    Read More

  4. 4th Grade Mental Math on Factors and Multiples |Worksheet with Answers

    Apr 16, 24 01:15 AM

    In 4th grade mental math on factors and multiples students can practice different questions on prime numbers, properties of prime numbers, factors, properties of factors, even numbers, odd numbers, pr…

    Read More

  5. Worksheet on Factors and Multiples | Find the Missing Factors | Answer

    Apr 15, 24 11:30 PM

    Worksheet on Factors and Multiples
    Practice the questions given in the worksheet on factors and multiples. 1. Find out the even numbers. 27, 36, 48, 125, 360, 453, 518, 423, 54, 58, 917, 186, 423, 928, 358 2. Find out the odd numbers.

    Read More