**Selina Concise Mathematics Class 10 ICSE Solutions Loci (Locus and Its Constructions)**

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**Selina ICSE Solutions for Class 10 Maths Chapter 16 Loci (Locus and Its Constructions)**

**Exercise 16(A)**

**Question 1.**

**Solution:**

**Question 2.**

**Solution:**

**Question 3.**

**Solution:**

**Question 4.**

Construct a triangle ABC, in which AB = 4.2 cm, BC = 6.3 cm and AC = 5 cm. Draw perpendicular bisector of BC which meets AC at point D. Prove that D is equidistant from B and C.

**Solution:**

**Question 5.**

**Solution:**

**Question 6.**

**Solution:**

**Question 7.**

**Solution:**

**Question 8.**

In parallelogram ABCD, side AB is greater than side BC and P is a point in AC such that PB bisects angle B. Prove that P is equidistant from AB and BC.

**Solution:**

**Question 9.**

In triangle LMN, bisectors of interior angles at L and N intersect each other at point A.

Prove that –

i) point A is equidistant from all the three sides of the triangle.

ii) AM bisects angle LMN.

**Solution:**

**Question 10.**

**Solution:**

**Question 11.**

**Solution:**

**Question 12.**

**Solution:**

**Question 13.**

Draw a line AB = 6 cm. Draw the locus of all the points which are equidistant from A and B.

**Solution:**

**Question 14.**

**Solution:**

**Question 15.**

**Solution:**

**Question 16.**

**Solution:**

**Question 17.**

**Solution:**

**Question 18.**

**Solution:**

**Question 19.**

On a graph paper, draw lines x = 3 and y = -5. Now, on the same graph paper, draw the locus of the point which is equidistant from the given lines.

**Solution:**

**Question 20.**

On a graph paper, draw the line x = 6. Now, on the same graph paper, draw the locus of the point which moves in such a way that its distance from the given line is always equal to 3 units.

**Solution:**

**Exercise 16(B)**

**Question 1.**

Describe the locus of a point at a distance of 3 cm from a fixed point.

**Solution:**

**Question 2.**

Describe the locus of a point at a distance of 2 cm from a fixed line.

**Solution:**

**Question 3.**

Describe the locus of the centre of a wheel of a bicycle going straight along a level road.

**Solution:**

**Question 4.**

Describe the locus of the moving end of the minute hand of a clock.

**Solution:**

**Question 5.**

Describe the locus of a stone dropped from the top of a tower.

**Solution:**

**Question 6.**

Describe the locus of a runner, running around a circular track and always keeping a distance of 1.5 m from the inner edge.

**Solution:**

**Question 7.**

Describe the locus of the door handle as the door opens.

**Solution:**

**Question 8.**

Describe the locus of a point inside a circle and equidistant from two fixed points on the circumference of the circle.

**Solution:**

**Question 9.**

Describe the locus of the centers of all circles passing through two fixed points.

**Solution:**

**Question 10.**

Describe the locus of vertices of all isosceles triangles having a common base.

**Solution:**

**Question 11.**

Describe the locus of a point in space which is always at a distance of 4 cm from a fixed point.

**Solution:**

The locus of a point in space is the surface of the sphere whose centre is the fixed point and radius equal to 4 cm.

**Question 12.**

**Solution:**

**Question 13.**

Describe the locus of a point in rhombus ABCD, so that it is equidistant from

i) AB and BC

ii) B and D.

**Solution:**

**Question 14.**

The speed of sound is 332 meters per second. A gun is fired. Describe the locus of all the people on the Earth’s surface, who hear the sound exactly one second later.

**Solution:**

The locus of all the people on Earth’s surface is the circumference of a circle whose radius is 332 m and centre is the point where the gun is fired.

**Question 15.**

Describe:

i) The locus of points at distances less than 3 cm from a given point.

ii) The locus of points at distances greater than 4 cm from a given point.

iii) The locus of points at distances less than or equal to 2.5 cm from a given point.

iv) The locus of points at distances greater than or equal to 35 mm from a given point.

v)The locus of the centre of a given circle which rolls around the outside of a second circle and is always touching it.

vi) The locus of the centers of all circles that are tangent to both the arms of a given angle.

vii) The locus of the mid-points of all chords parallel to a given chord of a circle.

viii) The locus of points within a circle that are equidistant from the end points of a given chord.

**Solution:**

**Question 16.**

Sketch and describe the locus of the vertices of all triangles with a given base and a given altitude.

**Solution:**

**Question 17.**

**Solution:**

**Question 18.**

By actual drawing obtain the points equidistant from lines m and n and 6 cm from the point P, where P is 2 cm above m, m is parallel to n and m is 6 cm above n.

**Solution:**

**Question 19.**

A straight line AB is 8 cm long. Draw and describe the locus of a point which is:

i) always 4 cm from the line AB

ii) equidistant from A and B.

Mark the two points X and Y, which are 4 cm from AB and equidistant from A and B. Describe the figure AXBY.

**Solution:**

**Question 20.**

**Solution:**

**Question 21.**

Draw a triangle ABC in which AB = 6 cm, BC = 4.5 cm and AC = 5 cm. Draw and label:

i) the locus of the centers of all circles which touch AB and AC.

ii) the locus of the centers of all circles of radius 2 cm which touch AB.

Hence, construct the circle of radius 2 cm which touches AB and AC.

**Solution:**

**Question 22.**

**Solution:**

**Question 23.**

O is a fixed point. Point P moves along a fixed line AB. Q is a point on OP produced such that OP = PQ. Prove that the locus of point Q is a line parallel to AB.

**Solution:**

**Question 24.**

**Solution:**

**Question 25.**

Construct a triangle ABC, with AB = 5.6 cm, AC = BC = 9.2 cm. Find the points equidistant from AB and AC; and also 2 cm from BC. Measure the distance between the two points obtained.

**Solution:**

**Question 26.**

Construct a triangle ABC, with AB = 6 cm, AC = BC = 9 cm. Find a point 4 cm from A and equidistant from B and C.

**Solution:**

**Question 27.**

**Solution:**

**Question 28.**

State the locus of a point in a rhombus ABCD, which is equidistant

i) from AB and AD;

ii) from the vertices A and C.

**Solution:**

**Question 29.**

Use a graph paper for this question. Take 2 cm = 1 unit on both the axes.

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.

**Solution:**

**Question 30.**

Construct an isosceles triangle ABC such that AB = 6 cm, BC=AC=4cm. Bisect angle C internally and mark a point P on this bisector such that CP = 5cm. Find the points Q and R which are 5 cm from P and also 5 cm from the line AB.

**Solution:**

**Question 31.**

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 lengths 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.

**Solution:**

**Question 32.**

Plot the points A(2,9), B(-1,3) and C(6,3) on a graph paper. On the same graph paper, draw the locus of point A so that the area of triangle ABC remains the same as A moves.

**Solution:**

**Question 33.**

**Solution:**

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