ICSE Class 10 Maths Chapter 16 Loci

Loci

Locus and Its Constructions

16.1 Locus

Locus is a Latin word from which the words location, locality, etc., are derived.

16.2 Definition

Locus is the path traced by a moving point, which moves so as to satisfy the certain given condition/conditions.

Example 1

Two parallel lines l and s are 4 cm apart. Find the locus of a point which is always equidistant from both the given lines.

Solution

Condition: The moving point is always equidistant from the given parallel lines l and s.

(i) As the distance between the given parallel lines is 4 cm and moving point is equidistant from these lines, so mark some points each \(\frac{4}{2} = 2\) cm from l and s. [see fig. (i)]

(ii) On joining all the points marked, a straight line AB is obtained which is the required locus. [see fig. (ii)]

From the final figure obtained, the required locus is the line AB which is parallel to both the given lines l and s and is also equidistant from both the lines.

The plural of locus is loci (pronounced as losai)

Example 2

Show that the locus of a point equidistant from a fixed point is a circle with the fixed point as centre.

Solution

Let O be the fixed point and we have to find the locus of a moving point P which moves in such a way that the distance between the moving point P and the fixed point O is always the same.

If the distance between the moving point P and the fixed point O is r cm, mark some points A, B, C, D, E, ..., etc, each at a distance of r cm from the fixed point O.

Now draw a free-hand curve through the marked points A, B, C, D, ..., etc.

We shall find that the final figure obtained is a circle with fixed point as centre and the distance between the moving point and the fixed point as radius.

Thus, the locus of a point equidistant from a fixed point is a circle with the fixed point as centre.

Teacher's Note

Understanding locus helps in navigation systems and GPS technology, where a point's path is tracked based on specific distance or angle conditions from reference points.

16.3 Theorems Based on Symmetry

Theorem 3

The locus of a point equidistant from two intersecting lines is the bisector of the angles between the lines.

Given: Two straight lines AB and CD intersecting at O. A point P is the interior of angle AOC such that it is equidistant from AB and CD.

To Prove: Locus of P is the bisector of angle AOC.

i.e. (i) P lies on bisector of angle AOC, and conversely.

(ii) every other point on the bisector of ∠AOC is equidistant from the intersecting lines AB and CD.

Construction: Draw a line through O and P. Then draw PL perpendicular to AB and PM perpendicular to CD.

(i) Proof

Therefore, P lies on the bisector of angle AOC.

(ii) Conversely

Let Q be any point on the bisector OP. Now to show that Q is equidistant from AB and CD, draw QR and QS perpendiculars to AB and CD respectively.

Clearly, \(\triangle\) OQR \(\cong\) \(\triangle\) OQS [By A.A.S. or A.S.A.]

\(\Rightarrow\) QR = QS [C.P.C.T.C.]

\(\Rightarrow\) Q is equidistant from AB and CD

The same results can be proved by taking a point, in the interior of angle COB or in the interior of angle AOD, etc.

Hence the theorem is proved.

Teacher's Note

Angle bisectors are used in architectural design and engineering to find points that are equally distant from two surfaces or walls, which is essential for symmetrical construction.

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