Loci Class 10 RS Aggarwal: ICSE Maths Goyal Brothers Prakashan Chapter-17

Loci Class 10 RS Aggarwal: ICSE Maths Goyal Brothers Prakashan Chapter-17

Que-1: Describe and construct each of the following loci :

[i]     The locus of the tip of a minute hand of a watch.

Ans: The locus of the moving tip of the minute hand of the clock will be a circle where radius will be the length of the minute hand.

[ii]     The locus of the tip of the pendulum of a clock.

Ans: According to figure, Pendulum of clock will be moving so by inspection, locus of end of pendulum will be an arc of circle

[iii]    The locus of a point 5 cm from a fixed point O.

Ans: Consider a point O and another point P where OP = 5 cm.Taking O as the centre and radius equal to OP, construct a circle. Hence, this circle is the locus of point P, which is at a distance of 5 cm from the given point O

[iv]    The locus of a point at a distance of 3 cm from a fixed line AB.

Ans: A pair of line each parallel to AB and a distance of 3 cm from AB.

[v]     The locus of a point equidistance from the arms OA and OB of angle AOB.

Ans: Bisector of angle AOP

[vi]    The locus of the centres of all circles, each of radius 1 cm and touching externally a fixed circles with centre O and radius 3 cm.

Ans: A circle with centre O and radius 4 cm

[vii]   The locus of the centres of all circles to which both the arms of an angle AOB are tangents.

Ans: Bisector of angle AOB

[viii]  The locus of a point 1 cm from the circumferences of a fixed circle towards the centre O, whose radius is 3 cm.

Ans: A concentric circle with center O and radius 2 cm.

[ix]    The locus of a point 1 cm from the centre of a circle of radius 2.5 cm.

Ans: A circle of radius 1 cmand concentric with the given circle.

[x]     The locus of a stone dropped from a tower.

Ans: A verticla straight line.

[xi]    AB is a fixed line. State the locus of a point P such that angle APB = 90

Ans: A circle with AB as diameter.

Que-2: Describe the locus of a point of a in a rhombus ABCD which is equidistant  from [i] AB and AD   [ii] A and C.

Ans:  (i) We know that,

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

∴ Locus of point equidistant from AB and AD is the angle bisector of ∠A.

In a rhombus diagonals bisect interior angles.

Hence, locus of point equidistant from AB and AD is diagonal AC.

(ii) We know that, The locus of a point which is equidistant from two fixed points is the perpendicular bisector of line segment joining the points.

∴ Locus of point equidistant from A and C is the perpendicular bisector of AC.

In a rhombus diagonals bisect each other at right angles.

Hence, locus of point equidistant from A and C is diagonal BD.

Que-3: Construct a angle ABC in which BC = 5.3 cm , CA = 4.8 cm and AB = 4 cm. find by construction a point P which is equidistant from BC and AB and also equidistant from B and C.

Que-4:    Construct angle AOB = 60. Mark a point P equidistant from OA and OB such that its distance from another given line CD is 3 cm.

Que-5:  Tringle ABC is an equilateral triangle of side 3.2 cm. Find the point on AB and AB produced which are 2 cm from BC.

Que-6:  Using ruler and compresses only, construct a angle ABC such that AB = 4.6 cm, BC = 3 cm and angle ABC = 30. Complete the rhombus ABDE such that C is equidistant from AB and BD. Locate the point Q on BC such that Q is equidistant from A and B.

Que-7:  AB and CD are two interesting lines. Find the position of the point distance 2 cm from AB and 1.8 cm from CD.

Que-8:  Using ruler and compasses only , construct a rhombus ABCD whose diagonals AC and BD are 8 cm and 6 cm long respectively. Find by construction , a point P equidistant from AB, AD and also equidistant from C and D. Measure PC.

Que-9:  Using ruler and compasses only, construct a angle ABC in which AB=4 cm ,BC =5 cm and angle  ABC =120.

[i] Locate the point p such that  angle  BAP` = 90 and = CP.

[ii]Measure the length of BP.

Que-10:  Using ruler and compass only , construct a angle ABC in which AB = 6 cm , BC = 3.5 cm and  CA = 4.6cm.

[i] Draw the locus of a point P which moves so that  it is always 3 cm from B.

[ii] Draw a locus of a point which moves so that it is equidistant from BC and CA .

[iii] Mark the point of intersection of the two loci obtained above . Measure PC.

Using ruler and compass only , construct an isosceles angle ABC in which BC= 5 cm ,AB =AC and angle  BAC = 90 .

[i]    P is equidistant from BC and AC .

[ii]   P is equidistant from B and C.

Que-11:  Using  ruler and compasses only ,construct a quadrilateral ABCD in which AB = 6cm,BC= 5cm ,          angle B = 60 , AD = 5 cm and D ia equidistant from AB and BC .

Que-12:  Using ruler and compasses only, construct a parallelogram ABCD in which AB =5.1cm , diagonal AC = 5.6 cm  and diagonal  BD = 7 cm .Locate point P on DC , which is equivalent from AB and BC.

Que-13:  Draw a circle of radius 4 cm and mark two cords AB and AC of the circle of length 6cm and 5 cm respectively .

[i] Construct the locus of point inside the circle , that are equivalent  from A and C .

[ii]  Construct the locus of  point  , inside the circle , that are equidistant from AB and AC.

Que-14:  A and B are fixed points 5 cm apart . The locus of the point P is the set of those point for which AP = 4 cm and locus of Q is the set of those points for which the BQ= 3.5 cm .

Que-15:  Construct the loci of P and Q and the points of intersection of the two loci , How many such loci points are there ?

Que-16: Using only a ruler and compasses , construct angle ABC = 120 , where AB =BC = 5cm .

(a) Mark two points D and E which satisfy the condition that they are equidistant from both BA and BC.

(b) In the above figure , join AE and  EC . Describe the figure .

(i)  ABCD       (ii) BD      (iii) ABE

 

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