**Exercise 14**

**1. A point moves such that its distance from a fixed line AB is always the same. What is the relation between AB and the path travelled by P?**

**Solution:**

**3. P is a fixed point and a point Q moves such that the distance PQ is constant, what is the locus of the path traced out by the point Q?**

**Solution:**

**4. (i) AB is a fixed line. State the locus of the point P so that ∠APB = 90°.**

**(ii) A,B are fixed points. State the locus of the point P so that ∠APB = 90°.**

**Solution:**

**5. Draw and describe the locus in each of the following cases:**

**(i) The locus of points at a distance 2.5 cm from a fixed line.**

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

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

**(iv) The locus of centres of all circles passing through two fixed points.**

**(v) The locus of a point in rhombus ABCD which is equidistant from AB and AD.**

**(vi) The locus of a point in the rhombus ABCD which is equidistant from points A and C.**

**Solution:**

_{1},C

_{2}and C

_{3}as the centres of circle which pass through A and B which are the two fixed points.

_{1},C

_{2}and C

_{3}.

**6. Describe completely the locus of points in each of the following cases:**

**(i) mid-point of radii of a circle.**

**(ii) centre of a ball, rolling along a straight line on a level floor.**

**(iii) point in a plane equidistant from a given line.**

**(iv) point in a plane, at a constant distance of 5 cm from a fixed point (in the plane).**

**(v) centre of a circle of varying radius and touching two arms of ∠ADC.**

**(vi) centre of a circle of varying radius and touching a fixed circle, centre O, at a fixed point A on it.**

**(vii) centre of a circle of radius 2 cm and touching a fixed circle of radius 3 cm with centre O.**

**Solution:**

**7. Using ruler and compasses construct:**

**(i) a triangle ABC in which AB = 5.5 cm,BC = 3.4 cm and CA = 4.9 cm.**

**(ii) the locus of points equidistant from A and C.**

**Solution:**

**8. Construct 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 and**

**(ii) P is equidistant from AB and BC**

**(iii) Measure and record the length of PB.**

**Solution:**

**9. A straight line AB is 8 cm long. Locate by construction the locus of a point which is:**

**(i) Equidistant from A and B.**

**(ii) Always 4 cm from the line AB.**

**(iii) Mark two points X and Y, which are 4 cm from AB and equidistant from A and B.**

**Name the figure AXBY.**

**Solution:**

**10. Use ruler and compasses only for this question.**

**(i) Construct Δ ABC, 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.**

**Solution:**

**11. Construct a triangle ABC with AB = 5.5 cm,AC = 6 cm and ∠BAC = 105°. Hence:**

**(i) Construct the locus of points equidistant from BA and BC.**

**(ii) Construct the locus of points equidistant from B and C.**

**(iii) Mark the point which satisfies the above two loci as P. Measure and write the length of PC.**

**Solution:**

**12. In the diagram, A,B and C are fixed collinear points; D is a fixed point outside the line. Locate:**

**(i) the point P on AB such that CP = DP.**

**(ii) the points Q such that CQ = DQ = 3 cm. How many such points are possible?**

**(iii) the points R on AB such that DR = 4 cm. How many such points are possible?**

**(iv) the points S such that CS = DS and S is 4 cm away from the line CD. How many such points are possible?**

**(v) Are the points P,Q,R collinear?**

**(vi) Are the points P,Q,S collinear?**

**Solution:**

**13. Points A,B and C represent position of three towers such that AB = 60 mm,BC = 73 mm and CA = 52 mm. Taking a scale of 10 m to 1 cm, make an accurate drawing of Δ ABC. Find by drawing, the location of a point which is equidistant from A,B and C and its actual distance from any of the towers.**

**Solution:**

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

**Solution:**

**15. Without using set square or protractor, construct the quadrilateral ABCD in which ∠BAD = 45°,AD = AB = 6 cm,BC = 3.6 cm and CD = 5 cm.**

**(i) Measure ∠BCD.**

**(ii) Locate the point P on BD which is equidistant from BC and CD.**

**Solution:**

**16. Without using set square or protractor, construct rhombus ABCD with sides of length 4 cm and diagonal AC of length 5 cm. Measure ∠ABC. Find the point R on AD such that RB = RC. Measure the length of AR.**

**Solution:**

**17. Without using set-squares or protractor construct:**

**(i) Triangle ABC, in which AB = 5.5 cm,BC = 3.2 cm and CA = 4.8 cm.**

**(ii) Draw the locus of a point which moves so that it is always 2.5 cm from B.**

**(iii) Draw the locus of a point which moves so that it is equidistant from the sides BC and CA.**

**(iv) Mark the point of intersection of the loci with the letter P and measure PC.**

**Solution:**

**18. By using ruler and compasses only, construct an isosceles triangle ABC in which BC = 5 cm,AB = AC and ∠BAC = 90°. Locate the point P such that:**

**(i) P is equidistant from the sides BC and AC.**

**(ii) P is equidistant from the points B and C.**

**Solution:**

**19. Using ruler and compasses only, construct a quadrilateral ABCD in which AB = 6 cm,BC = 5 cm,∠B = 60°,AD = 5 cm and D is equidistant from AB and BC. Measure CD.**

**Solution:**

**20. Construct an isosceles triangle ABC such that AB = 6 cm,BC = AC = 4 cm. Bisect ∠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.**

**Solution:**

**21. 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 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:**

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

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

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

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

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

**Solution:**

**Chapter-Test**

**1. Draw a straight line AB of length 8 cm. Draw the locus of all points which are equidistant from A and B. Prove your statement.**

**Solution:**

**2. A point P is allowed to travel in space. State the locus of P so that it always remains at a constant distance from a fixed point C.**

**Solution:**

**3. Draw a line segment AB of length 7 cm. Construct the locus of a point P such that area of triangle PAB is 14 cm**

^{2}.**Solution:**

^{2}

^{2}

**4. Draw a line segment AB of length 12 cm. Mark M, the mid-point of AB. Draw and describe the locus of a point which is**

**(i) at a distance of 3 cm from AB.**

**(ii) at a distance of 5 cm from the point M. Mark the points P,Q,R,S which satisfy both the above conditions. What kind of quadrilateral is PQRS? Compute the area of the quadrilateral PQRS.**

**Solution:**

^{2}

**5. AB and CD are two intersecting lines. Find the position of a point which is at a distance of 2 cm from AB and 1.6 cm from CD.**

**Solution:**

**6. Two straight lines PQ and PK cross each other at P at an angle of 75°. S is a stone on the road PQ,800 m from P towards Q. By drawing a figure to scale 1 cm = 100 m, locate the position of a flagstaff X, which is equidistant from P and S, and is also equidistant from the road.**

**Solution:**

**7. Construct a rhombus PQRS whose diagonals PR,QS are 8 cm and 6 cm respectively. Find by construction a point X equidistant from PQ,PS and equidistant from R,S. Measure XR.**

**Solution:**

**8. Without using set square or protractor, construct the parallelogram ABCD in which AB = 5.1 cm, the diagonal AC = 5.6 cm and diagonal BD = 7 cm. Locate the point P on DC, which is equidistant from AB and BC.**

**Solution:**

**9. By using ruler and compass only, construct a quadrilateral ABCD in which AB = 6.5 cm,AD = 4 cm and ∠DAB = 75°. C is equidistant to from the sides if AB and AD, if also C is equidistant from the points A and B.**

**Solution:**