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.
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
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.
Recent Articles
-
Quarter Past and Quarter To | Quarter Past Hour | Quarter to Next Hour
Nov 23, 24 03:45 PM
The hands of clock move from left to right. This is called the clock wise motion. When the minute hand is on the right side of the clock, it shows the number of minutes past the hour. When the minute… -
Half Past an Hour | What does Half Past Mean? | Half an Hour|Half Past
Nov 23, 24 03:14 PM
We learnt that, one hour is equal to 60 minutes. When one hour is divided into two, it is half an hour or 30 minutes. The minute hand points at 6. We say, 30 minutes past an hour or half past an hour… -
Telling the Time | Teaching Time | Analogue Clock| Reading Time
Nov 23, 24 02:51 PM
Teaching time is an interactive activity for telling time. This activity helps students to learn how to read the clock to tell time using the analogue clock. While reading or observing the time on a -
2nd Grade Fractions Worksheet | Basic Concept of Fractions | Answers
Nov 23, 24 12:22 AM
In 2nd Grade Fractions Worksheet we will solve different types of problems on fractions, one-whole, one-half, one-third, one-fourth, three-fourth or s quarter. In a fraction, it is important that the… -
Time Duration |How to Calculate the Time Duration (in Hours & Minutes)
Nov 22, 24 12:34 AM
Time duration tells us how long it takes for an activity to complete. We will learn how to calculate the time duration in minutes and in hours. Time Duration (in minutes) Ron and Clara play badminton…
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.