# Locus Concise Maths Solutions Chapter-16 ICSE Class 10

## Chapter-16 Loci (Locus and Its Constructions)

**Locus Concise Maths Solutions** Chapter-16 ICSE Class 10.** Solutions** of Exercise – 16 (A), Exercise – 16 (B), for** Concise **Selina Maths of ICSE Board Class 10th. **Concise Solutions Reflection **Chapter – 12 for ICSE Maths Class 10 is available here. All **Solutions **of **Concise** Selina of Chapter-16 **Locus** has been solved according instruction given by council. This is the **Solutions **of Chapter-16 **Locus** for ICSE Class 10th. ICSE Maths text book of **Concise** is In series of famous ICSE writer in maths publications. **Concise** is most famous among students

## Locus Concise Maths Solutions Chapter-16 ICSE Class 10

The **Solutions** of **Concise** Mathematics Chapter-16 **Locus** for ICSE Class 10 have been solved. Experience teachers Solved Chapter-16 **Locus** to help students of class 10th ICSE board. Therefore the ICSE Class 10th **Maths** **Solutions** of **Concise** Selina Publishers helpful on various topics which are prescribed in most ICSE Maths textbook

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** Exercise- 16(A) , Exercise- 16(B)**

**How to Solve Concise Maths Selina Publications Chapter-16 Locus**

Note:- Before viewing **Solutions** of Chapter-16 **Locus**** **of **Concise **Selina **Maths** read the Chapter Carefully then solve all example of your text book**. **The Chapter-16 **Locus**** **is main Chapter in ICSE board

Locus Concise Maths Solutions Chapter-16 ICSE Class 10 **Exe- 16-(A)**

**Question 1.**

**Given— PQ is perpendicular bisector of side AB of the triangle ABC**

Prove – Q…………………B

**Answer 1**

**Question 2**

**Given— CP is bisector of angle C of A ABC.**

**Prove: p is equidistant from AC and BC**

**Answer 2**

**Prove: p is equidistant from AC and BC**

**Solution:**

**Question 3.**

**Given— AX bisects angle BAG and PQ is perpendicular bisector of AC which meets AX at point Y.**

**Prove:**

(i) X is equidistant from AB and AC.

(ii) Y is equidistant from A and C.

**Answer 3**

**Construction:** From X, draw XL ⊥ AC and XM ⊥ AB and join YC

**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.**

**Answer 4**

**Given:** In Δ ABC, AB, = 4.2 cm, BC = 6.3 cm and AC = 5cm

**Question 5.**

**In each of the given figures: PA = PH and QA = QB.**

**Prove, in each case, that PQ (produce, if required) is perpendicular bisector of AB. Hence, state the locus of points equidistant from two given fixed points.**

**Answer 5**

**(i)** **Construction:** Join PQ which meets AB in D.

**Question 6****.**

**Construct a right angled triangle PQR, in which ∠Q = 90°, hypotenuse PR = 8 cm and QR = 4.5 cm. Draw bisector of angle PQR and let it meets PR at point T. Prove that T is equidistant from PQ and QR.**

**Answer 6**

**Question 7.**

**Construct a triangle ABC in which angle ABC = 75°. AB = 5 cm and BC = 6.4 cm.**

**Draw perpendicular bisector of side BC and also the bisector of angle ACB. If these bisectors intersect each other at point P ; prove that P is equidistant from B and C ; and also from AC and BC.**

**Answer 7**

**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.**

**Answer 8**

**Given: ** In || gm ABCD. AB > BC and bisector of ∠B meets diagonal AC at P.

**To Prove:** P is equidistant from AB and BC.

**Construction:** From P, draw PL ⊥ AB and PM ⊥ BC.

**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.**

**Answer 9**

**Question 10.**

**Use ruler and compasses only for this question:**

**(i) construct ΔABC, where AB = 3.5 cm, BC = 6 cm and ∠ABC = 60°.**

**(ii) Construct the locus of points 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 the point P which is equidistant from AB, BC and also equidistant from B and C. Measure and record the length of PB. (2010)**

**Answer 10**

**Steps of construction:**

1. Draw a line BC = 6 cm and an angle CBX = 60°. Cut off AB = 3.5 cm. Join AC, ΔABC is the required triangle.

2. Draw ⊥ bisector of BC and bisector of ∠B.

3. Bisector of ∠B meets bisector of BC at P

∴ BP is the required length, where PB = 3.5 cm

4. P is the point which is equidistant from BA and BC, also equidistant fromB and C

**Question 11.**

**The given figure shows a triangle ABC in which AD bisects angle BAC. EG is perpendicular bisector of side AB which intersects AD at point E Prove that:**

**(i) F is equidistant from A and B.**

**(ii) F is equidistant from AB and AC.**

**Answer 11**

**Given :** In the figure,

In ΔABC, AD is the bisector of ∠BAC Which meets BC at D EG is the perpendicular bisector of AB which intersects AD at F

**To prove :**

(i) F is equidistant from A and B.

(ii) F is equidistant from AB and AC.

**Proof:**

(i) ∴ F lies on the perpendicular bisector of AB F is equidistant from A and B

(ii) Again,

∴ F lies onthe bisector of ∠BAC

∴ F is equidistant from AB and AC.

(10 cm theorem)

Hence proved.

**Question 12.**

**The bisectors of ∠B and ∠C of a quadrilateral ABCD intersect each other at point P. Show that P is equidistant from the opposite sides AB and CD.**

**Answer 12**

In quadrilateral ABCD, the bisectors of ∠B and ∠C meet each other at P.

**Question 13.**

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

**Answer 13**

**Steps of Construction:**

(i) Draw a line segment AB = 6 cm

(ii) Draw perpendicular bisector LM of AB. LM is the required locus.

(iii) Take any point on LM say P.

(iv) Join PA and PB

∵ P lies on the right bisector of line AB

**Question 14.**

**Draw an angle ABC = 75°. Draw the locus of all the points equidistant from AB and BC.**

**Answer 14**

**Steps of Construction:
(I) Draw a ray BC.
(ii) Construct a ray RA making an angle of 750 with BC.
(iii) ∴ ∠ABC = 75°
(iv) Draw the angle bisector BP of ∠ABC. BP is the required locus.
(v) Take any point D on BP.
(vi) From D, draw DE ⊥ AB and DF ⊥ BC.
∵ D lies on the angle bisector ∠ABC.
∴ D is equidistant from AB and BC.
∴ DE = DF
Similarly any point on BP, is equidistant from AB and BC.
∴BP is the locus of all points which are equidistant from AB and BC.
**

**Question 15.**

**Draw an angle ABC = 60°, having AB = 4.6 cm and BC = 5 cm. Find a point P equidistant from AB and BC ; and also equidistant from A and B**

**Answer 15**

**Steps of Construction:
(i) Draw a line segment BC = 5 cm.
(ii) At B, draw a ray BX making an angle of 60° and cut off BA = 4.6 cm.
(iii) Draw the angle bisector of ∠ABC.
(iv) Draw the perpendicular bisector of AB which intersects the angle bisector at P.
P is the required point which is equidistant from AB and BC as well as from points A and B.**

**Question 16.**

**In the figure given below, find a point P on CD equidistant from points A and B.**

**Answer 16**

** **

**Steps of Construction:
(i) In the figure AB and CD are two line segments.
(ii) Draw the perpendicular bisector of AB which intersects CD in P.
P is the required point which is equidistant from A and B
∵ P lies on the perpendicular bisector of AB.
∴ PA = PB.**

**Question 17.**

**In the given triangle ABC, find a point P equidistant from AB and AC; and also equidistant from B and C.**

**Answer 17**

** **

** Steps of Construction:
(i) In the given triangle, draw the angle bisector of ∠BAC.
(ii) Draw the perpendicular bisector of BC which intersects the angle bisector of ∠A at P.
P is the required point which is equidistant from AB and AC as well as from B and C.
∵ P lies on the angle bisector of ∠BAC.
∴ It is equidistant from AB and AC. Again
∵ P lies on the perpendicular bisector of BC.
∴ P is equidistant from B and C.**

**Question 18.**

**Construct a triangle ABC, with AB = 7 cm, BC = 8 cm and ∠ABC = 60°. Locate by construction the point P such that :
(i) P is equidistant from B and C.
(ii) P is equidistant from AB and BC.
(iii) Measure and record the length of PB.
(2000)
**

**Answer 18**

**Steps of Construction :
1. Draw a line segment AB = 7 cm.
2. Draw angle ∠ABC = 60° with the help of compass.
3. Cut off BC = 8 cm.
4. Join A and C.
5. The triangle ABC so formed is required triangle.
(i) Draw perpendicular bisector of line BC. The point situated on this line will be equidistant from B and C.
(ii) Draw angular bisector of ∠ABC. Any
point situated on this angular bisector is equidistant from lines AB and BC.**

** **

** **

**The point which fulfills the condition required in (i) and (ii) is the intersection point of bisector of line BC and angular bisector of ∠ABC.
(iii) Length of PB is 4.5 cm.**

**Question 19.**

**On a graph paper, draw the 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.
**

**Answer 19**

**On the graph paper, draw axis XOX’ and YOY’ Draw a line l, x = 3 which is parallel to y-axis and another line m, y = -5, which is parallel to x-axis**

**These two lines intersect eachother at P.
Now draw the angle bisector p of ∠P
∵ p is the bisector of ∠P
∴ Any point on P, is equidistant from l and m
∴ This line p is equidistant from l and m.**

**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.**

**Answer 20**

** **

**On the graph, draw axis XOX’ and YOY’
Draw a line l, x = 6
Which is parallel to -axis
Take point P and Q which are at a distance of 3units from the line l
Draw line rn and n from P and Q parallel to P respectively
The line m and n are the required locus of the points P and Q
Which arc always 3 units from the line l.
Hence proved.**

** ****EXERCISE -16 B**, Concise Maths Solutions Chapter-16 Locus

** **

**Question 1.**

**Describe the locus for questions 1 to 13 given below: 1. The locus of a point at a distance 3 cm from a fixed point**

**Answer 1**

The locus of a point which is 3 cm away from a fixed point is circumference of a circle whose radius is 3 cm and the fixed point is called the centre of the circle.

**Question 2.**

**The locus of points at a distance 2 cm from a fixed line.**

**Answer 2**

A pair of straight lines 1 and m which are parallel to the given line at a distance of 2 cm, from it is the locus.

**Question 3.**

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

**Answer 3**

The locus of the centre of a wheel which is going straight along a level road will be a straight line parallel to the road at a distance equal to the radius of the wheel.

**Question 4.**

**The locus of the moving end of the minute hand of a clock.**

**Answer 4**

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

**Question 5.**

**The locus of a stone dropped from the top of a tower.**

**Answer 5**

The locus of stone which is dropped from the top of the tower will be a vertical line through the point from which the stone is dropped.

**Question 6.**

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

**Answer 6**

The locus of the runner running round a circular track at a distance of 1.5 m from the inner edge will be the circum¬ference of a circle whose radius is equal to the radius of the inner circular track plus 1.5 m

**Question 7.**

**The locus of the door-handle as the door opens.**

**Answer 7**

The locus of the door handle will be the circumfer-ence of a circle with centre at the axis of rotation of the door and radius equal to the distance between the door handle and the axis of rotation of the door.

**Question 8.**

**The locus of points inside a circle and equidistant from two fixed points on the circumference of the circle.**

**Answer 8**

The locus of the points inside the circle which are equidistant from the fixed points on the circumference of the circle will be the diameter which is perpendicular bisector of the line joining the two fixed points on the circle.

**Question 9.**

**The locus of the centres of all circles passing through two fixed points.**

**Answer 9**

The locus of the centre of all the circles which pass through two fixed points will be the perpendicular bisector of the line segment joining the two fixed points which are given.

**Question 10.**

**The locus of vertices of all isosceles triangles having a common base.**

**Answer 10**

The locus of vertices of all isosceles triangles have a common base will be the perpendicular bisector of the common base of the triangles.

**Question 11.**

**The locus of a point in space, which is always at a distance of 4 cm from a fixed point.**

**Answer 11**

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.**

**The locus of a point P, so that:**

**AB ^{2} = AP^{2}+ BP^{2}, where A and B are two fixed points.**

**Answer 12**

The locus of the point P is the circumference of a circle with AB as diameter and satisfies the condition AB^{2} = AP^{2}+ BP^{2}.

**Question 13****.**

**The locus of a point in rhombus ABCD, so that it is equidistant from**

**(i) AB and BC ; (ii) B and D.**

**Answer 13**

Locus of the point in a rhombus ABCD which is equidistant from.

**Question 14.**

**The speed of sound is 332 metres per second. A gun is fired. Describe the locus of all the people on the earth’s surface, who hear the sound exactly after one second.**

**Answer 14**

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 centres of all circles that are tangent to both the arms of a given angle.**

**(vii) The locus of the mid-points of all chords par-allel 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.**

**Answer 15**

(i) The space inside of the circle whose radius is 3 cm and centre is the fixed point which is given.

(ii) The space outside of the circle whose radius is 4 cm and centre is the fixed point which is given.

(iii) The space inside and circumference of the circle with a raduis of 2.5 cm and centre is the given fixed point.

(iv) The space outside and the circumference of a circle with a radius of 35 mm and centre is the given fixed point.

(v) Circumference of the circle concentric with the second circle whose radius is equal to the sum of the radii of the two given circles.

(vi) The locus of the centre of all circle whose tan-gents are the arms of a given angle is the bisector of that angle.

(vii) The locus of the mid-points of the chords which are parallel to a given chords is the diameter perpendicular to the given circle.

The locus of points within a circle that are equidistant from the end points of a given chord is a diameter which is perpendicular bisector of the given chord

**Question 16.**

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

**Answer 16**

**Steps of Construction:**

Draw a line XY parallel to the base BC from the vertex A.

This line is the locus of vertex A. All the tri-angles which have the base BC and altitude (length) equal to AD.

**Question 17.**

**In the given figure, obtain all the points equidistant from lines m and n ; and 2.5 cm from O.**

**Answer 17**

Draw angle bisector PQ and XY of angles formed by the lines m and n. From O, draw arcs with radius 2.5 cm, which intersects the angle bisectors at A, B, C and D respectively.

Hence A, B, C and D are the required points.

**P.Q.**

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

**Answer 17**

**Steps of construction:**

(i) Draw a line n.

(ii) Take a point P on n and draw a perpendicular to n. ,

(iii) Cut off LM = 6 cm and draw a line q, the per pendicu lar bisector of LM

(iv) At M, draw a line n making an angle of 900 at.

(v) Produce LM and take a point P such that PM =2 cm.

(vi) From P, draw are an with 6 cm radius which intersects the line q, (he perpendicular bisector cf LM, atA and B.

A and B are the required points which are equidisant from rn and n and is at a distance of 6 cm from p

**Question 18.**

**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.**

**Answer `18**

**Steps of Construction:**

(i) Draw a line segment AB equal to 8 cm.

(ii) Draw two parallel lines ℓ and m to AB at a distance of 4cm

(iii) Draw the perpendicular bisector of AB which intersects the parallel lines ℓ and m at X and Y respectively then X and Y are the required points.

(iv) Join AX.AY, BX and BY.

The figure so formed is a square as its diago¬nals are equal and intersect at 90°.

**Question 19.**

**Angle ABC = 60° and BA = BC = 8 cm. The mid-points of BA and BC are M and N respec¬tively. Draw and describe the locus of a point which is :**

**(i) equidistant from BA and BC.**

**(ii) 4 cm from M.**

**(iii) 4 cm from N.**

**Mark the point P, which is 4 cm from both M and N, and equidistant from BA and BC. Join MP and NP, and describe the figure BMPN.**

**Answer 19**

(i) Draw an angle of 60° with AB = BC = 8 cm.

(ii) Draw the angle bisector BX of ∠ABC.

(iii) With centre M and N, draw circles of radius equal to 4 cm, which intersect each other at P. P is the required point.

(iv) Join MP, NP.

BMPN is a rhombus.

∵ MP = PM = BN = PN = 4 cm.

**Question 20.**

**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 centres of all circles which touch AB and AC ;**

**(ii) the locus of the centres of all the circles of radius 2 cm which touch AB.**

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

**Answer 20**

(i) Draw a line segment BC = 4.5 cm

(ii) With centre B and radius 6 cm and with centre C and radius 5 cm, draw arcs which intersects each other at point A

(iii) Join AB and AC.

ABC is a required triangle.

(iv) Draw the angle bisector of ∠BAC.

(v) Draw a lines parallel to AC and AB at a distance of 2 cm. which intersects each other AD at O.

(vi) With centre O and radius 2 cm, draw a circle which touches AB and AC.

**Question 21.**

**Construct a triangle ABC, having given AB = 4.8 cm. AC = 4 cm and ∠A = 75°. Find a point P.**

**(i) inside the triangle ABC.**

**(ii) outside the triangle ABC.**

**equidistant from B and C; and at a distance of 1.2 cm from BC.**

**Answer 21**

(i) Draw a line segment AB = 4.8 cm.

(ii) At, A draw a ray AX making an angle of 75°.

(iii) Cut off AC = 4 cm from AX.

(iv) Join BC.

(v) Draw two lines ℓ and m parallel to BC at a distance of 1.2 cm.

(vi) Draw the perpendicular bisector of BC which intersect ℓ and m at P and P .

P and P1 are the required points which are in¬side and outside the given triangle ABC.

**P.Q.**

**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**

**Answer 22**

O is fixed point. P moves along AB; a fixed line. OP is joined and produced it to Q such that OP = PQ, Now we have to prove that locus of P is a line parallel to AB.

**Proof:**

∵ P moves along AB, and Q moves in such a way that PQ is always equal to OP.

P is the mid-point of OQ.

Now is A OQQ’

P and P’ are the mid-point of OQ and OQ’

AB || OQ’

Locus of Q is a line CD, which is parallel to AB.

**Question 22.**

**Draw an angle ABC = 75°. Find a point P such that P is at a distance of 2 cm from AB and 1.5 cm from BC.**

**Answer 22**

**Steps of Construction:**

(i) Draw a ray BC.

(ii) At B, draw a ray BA making an angle of 75° with BC.

(iii) Draw a line ℓ parallel to AB at a distance of 2 cm.

(iv) Draw another line m parallel to BC at a distance of 1.5 cm. which intersects m at P.

∴ P is the required point.

**Question 23.**

**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.**

**Answer 23**

(i) Draw a line segment AB = 5.6 cm.

(ii) From A and B, as centres and radius 9.2 cm, draw the arcs which intersect each other at C.

(iii) Join CA and CB.

(iv) Draw two lines m and n parallel to BC at a distance of 2 cm each.

(v) Draw the angle bisector of ∠CAB which inter-sects the parallel lines m and n at P and Q respectively.

P and Q are the required points which are equi-distant from AB and AC.

On measuring the distance between P and Q is 4.3 cm.

**Question 24.**

**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.**

**Answer 24**

Steps of Construction:

i) Draw a line segment AB = 6 cm

ii) With A and B as centers and radius 9 cm, draw two arcs which intersect each other at

(iii) Join CA and CB.

(iv) Draw the perpendicular bisector of BC.

(v) A as centre and radius 4 cm, draw an arc which intersect the perpendicular bisector of BC, at P.

P is the required point which is equidistant from B and C and at a distance of 4 cm from A.

**Question 25.**

**Ruler and compasses 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 a triangle ABC, in which BC = 6 cm, AB = 9 cm and angle ABC = 60°.**

**(ii) Construct the locus of all points inside triangle ABC, which are equidistant from B and C.**

**(iii) Construct the locus of the vertices of the triangles with BC as base and which are equal in area to triangle ABC.**

**(iv) Mark the point Q, in your construction, which would make A QBC equal in area to A ABC, and isosceles.**

**(v) Measure and record the length of CQ. [1998]**

**Answer 25**

(i) Draw a line segument BC = 6 cm.

(ii) At B, draw a ray BX making an angle of 60H and cut off BA = 9 cm.

(iii) Join AC, then A ABC is the given triangle. ..(i)

(iv) Draw perpendicular bisector of BC which

intersects BA in M, then any point on LM, is the equidistant from B and C.

(v) Though A, draw a line m II BC.

(vi) The perpendicular bisector of BC and the par-allel line intersect each other at Q.

(vii) Join QB and QC.

Then A QBC is equal in the area of A ABC and through any point on line m, and bace BC, every triangle is equal in area to the given triangle ABC. Length of CQ, on measuring.

**Question 26.**

**State the locus of a point in a rhombus ABCD, wi.r’ch is equidistant**

**(i) from AB and AD; (ii) from the vertices A and C.**

**[1998]**

**Answer 26**

In rhombus ABCD, draw the angle bisector of ∠A which meets in C

∴ Join BD, which intersects AC at O.

O is the required locus.

From O, draw OL ⊥ AB and OM ⊥ AD.

In Δ AOL and Δ AOM

∴ O is equidistant from AB and AD.

∵ Diagonal AC and BD bisect each other at O at right angles.

∴ AO = OC

O is equidistant from A and C.

**Question 27.**

**Use 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 ABandAC.**

**(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.**

**[1999]**

**Answer 27**

Plot the points A (1, 1), B (5, 3) and C (2, 7) on the graph and join AB, BC and CA.

Draw the perpendicular bisector of AB and angle bisector of ∠A which intersect each other at P. P is the required point,

∵ P lies on the perpendicular bisector of AB.

∴P is equidistant from A and B.

Again,

∵ P lies on the angle bisector of ∠A

∴ P is equidistant’from AB and AC

Now, on measuring the length of PA, it is 5.2 cm

(Approx.)

**Question 28.**

**Construct an isosceles triangle ABC such that AB = 6 cm, BC = AC = 4 cm. Biscet ∠C internally and mark a point P on this bisector such that CP = 5 cm. Find the points Q and R which are 5 cm from P and also 5 cm from the line AB. [2001]**

**Answer 28**

(i) Draw a line segment AB = 6 cm.

(ii) With centres A and B and radius 4 cm, draw two arcs which intersect each other at c

(iii) Join CA and CB.

(iv) Draw the angle bisector of ∠C and cut off CP – 5 cm.

(v) A line m is drawn parallel to AB at a distance of 5 cm.

(vi) P as centre and radius 5 cm, draw arcs which

intersect the line m at Q and R.

(vii) Join PQ, PR and AQ.

Q and R are the required points.

**Question **

**P.Q ****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. [1995]**

**Answer PQ**

(i) Draw a circle with radius = 4 cm and O is the centre.

(ii) Take a point A on it.

(iii) A as centre and radius 6 cm draw an arc which intersects the circle at B.

(iv) Again A as centre and radius 5 cm, draw another arc which intersects the circle at C.

(v) Join AB and AC.

(vi) Draw the perpendicular bisector of AC, which intersects AC at M and meets the circle at E and F. EF is the locus of the points inside the circle which are equidistant from A and C.

(vii) Join AE, AF, CE and CF.

Similarly, we can prove that CF = AF Hence EF is the locus of points which are equidistant from A and C.

(ii) Again draw the bisector of ∠A which meets the circle at N.

∴ Locus of points inside the circle which are equidistant from AB and AC is the perpendicular bisector of ∠A.

**Question 29.**

**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 ΔABC remains the same as A moves.**

**Answer 29**

Draw axis XOX’ and YOY’ on the graph paper. Plot the points A (2, 9), B (-1,3) and C (6, 3) Join AB, BC and CA which form a ΔABC. From A, draw a line l parallel to BC on x-axis The locus of point A is the line l.

∵ l || BC and triangles on the same base BC and between the same parallel are equal in area.

∴ l is the required locus of point A.

**Question 30.**

**Construct a triangle BCP given BC = 5 cm, BP = 4 cm and ∠PBC = 45°.**

**(i) Complete the rectangle ABCD such that:**

**(a) P is equidistant from A B and BC.**

**(b) P is equidistant from C and D.**

**(ii) Measure and record the length of AB.**

**(2007)**

**Answer 30 **

(i) Draw a line segment BC = 5 cm.

(ii) At B, draw a ray making an angle of 45° and cut off BP = 4 cm.

(iii) Join PC.

(iv) v P is equidistant from AB and BC.

∴ P lies on the bisector of ∠ABC.

Now draw a ray BY making an angle of 90°.

P is equidistant from C and D P lies on the perpendicular bisector of CD.

(v) From C, draw CZ ⊥ BC which intersect the perpendicular bisector at Q.

(vi) Cut off QD = CQ and from BP, cut off BA = CD.

(vii) Join AD.

Then ABCD is the required rectangle. Measuring the length of AB, it is 5.7 cm approximately.

**Question 31.**

**Use ruler and compasses only for the following questions. All constructions lines and arcs must be clearly shown.**

**(i) Construct a ΔABC in which BC = 6.5 cm, ∠ABC = 60°, AB = 5 cm.**

**(ii) Construct the locus of points at a distance of 3.5 cm from A.**

**(iii) Construct the locus of points equidistant from AC and BC.**

**(iv) Mark 2 points X and Y which are at**

**distance of 3.5 cm from A and also equidistant from AC and BC. Measure XY. (2016)**

**Answer 31**

**(i) Steps of construction :**

(1) Draw BC = 6.5 cm using a ruler.

(2) At B, draw ∠CBP = 60°

From BP, cut off BA = 5 cm.

(3) Join AC to get the required triangle.

(4) With A as a centre and radius equal to 3.5 cm, draw a circle. This circle is the required

locus of points at a distance of 3.5 cm from A.

(5) Draw the bisector of ∠ACB. This bisector is the locus of points equidistant from AC and BC.

(6) The angle bisector drawn above cuts the circle at X and Y. These are the points which are at a distance of 3.5 cm from A and also equidistant from AC and BC. On measuring, the length of XY comes out to be 5.2 cm

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