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Loci (Locus and its Constructions) — Mathematics STD 10 — Question

ICSE BoardEnglish MediumSTD 10MathematicsLoci (Locus and its Constructions)4 Marks
Question
Use ruler and compasses only for this question. Draw a circle of radius 4 cm and mark two chords AB and AC of the circle of length f 6 cm and 5 cm respectively.
(i) Construct the locus of points, inside the circle, that are equidistant from A and C. Prove your construction.
(ii) Construct the locus of points, inside the circle, that are equidistant from AB and AC.
✓

Answer

(I) Draw PQ, the perpendicular bisector of chord AC. PQ is the required locus, which is the diameter of the circle.
Reason: We know each point on the perpendicular bisector of AB is equidistant from A and B. Also the perpendicular bisector of a chord, passes through the centre of the circle and any chord passing through the centre of the circle is its diameter.

∴ PQ is the diameter of the circle.
(ii) Chords AB and AC intersects at M and N is a moving point such that LM = LN, where LM ⊥ AB and LN ⊥ AC
In right ΔALN and ΔALB
∠ANL = ∠ABL ...(90° each)
AL = AL ...(Common)
NL = BL ...[Given]
∴ ΔALN = ΔALB ...[R.H.S.]
Hence ∠MAL = ∠BAL ...c.p.c.t.
Thus, L lies on the bisector of ∠BAC.
Hence proved.

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