Loci (Locus and its Constructions) Mathematics - Loci (Locus and its Constructions)

In ΔABC, the bisector AX of ∠A intersects BC ar X. XL ⊥ AB and XM ⊥ AC are drawn (Fig.) is XL = XM? why or why not?

Prove that the common chord of two intersecting circles is bisected at right angles by the line of centres.

The bisector of ∠ B and ∠C of a quadrilateral ABCD intersect in P, show that P is equidistant from the opposite sides AB and CD.

Find the locus of the centre of a circle of radius r touching externally a circle of radius R.

Prove that the common chord of two intersecting circles is bisected at right angles by the line of centres.

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.

The bisectors of ∠B and ∠C of a quadrilateral ABCD intersect in P. Show that P is equidistant from the opposite sides AB and CD.

In Fig. AB = AC, BD and CE are the bisectors of ∠ABC and ∠ACB respectively such that BD and CE intersect each other at O. AO produced meets BC at F. Prove that AF is the right bisector of BC.

How will you find a point equidistant from three given points A, B, C which are not in the same straight line?

Draw two intersecting lines to include an angle of 30°. Use ruler and compasses to locate points which are equidistant from these Iines and also 2 cm away from their point of intersection. How many such points exist?

In Fig. ABCD is a quadrilateral in which AB = BC. E is the point of intersection of the right bisectors of AD and CD. Prove that BE bisects ∠ABC.

Use graph paper for this question. Take 2 cm = 1 unit on both the axis.
(i) Plot the points A(1,1), B(5,3) and C(2,7).
(ii) Construct the locus of points equidistant from A and B.
(iii) Construct the locus of points equidistant from AB and AC.
(iv) locate the point P such that PA = PB and P is equidistant from AB and AC.
(v) Measure and record the length PA in cm.

Given ∠BAC (Fig), determine the locus of a point which lies in the interior of ∠BAC and equidistant from two lines AB and AC.

Ruler and compass only may be used in this question. All construction lines and arcs must be clearly shown, and be of sufficient length and clarity to permit assessment.
(i) Construct Δ ABC, in which BC = 8 cm, AB = 5 cm, ∠ ABC = 60°.
(ii) Construct the locus of point inside the triangle which are equidistant from BA and BC.
(iii) Construct the locus of points inside the triangle which are equidistant from B and C.
(iv) Mark as P, the point which is equidistant from AB, BC and also equidistant from B and C.
(v) Measure and record the length of PB.