Topic 4 Loci

Key Words

loci

path

equidistant

By the end of this topic, you should be able to:

1. describe common types of loci.
2. construct loci involving points under given conditions.
3. construct intersecting loci.
4. construct loci involving inequalities.

Introduction

In Senior One, you looked at the introductory concepts of loci. Loci may involve a mental strategy of imagining yourself placing items around a room, such as on a couch, next to a lamp, or along a structured pathway in a neighbourhood. compound, garden or This topic will help you to understand and apply loci in different conditions.

4.1 Describing Common Types of Loci

Activity 4.1(a) (Work in groups)

(a) Study the dimensions of a certain farm shown below:

(b) Describe how you would take the path such that you are always:

i) 2 m away from A.

ii) 1.5 m away from the line joining A to B.

iii) the same distance from A as from C.

(c) What conclusion can you make from (b) above?

Note

From the Activity 4.1(a) above, you observe that in each case, you would not place the dot anywhere you like, since you had to obey certain conditions. This section will engage you in describing the common loci.

Activity 4.1(b) (Work in groups) 26

(a) Study the shape shown.

(b) Identify any point on the circumference of this locus and measure the distance from the centre to that point.

(c) Identify other points on the circumference and measure the distance from the centre to them.

(d) What conclusion can you make?

(e) Discuss the meaning of locus.

(f) Present your conclusion to the plenary.

Activity 4.1(c) (Work in groups) (a) Fold a sheet of paper at least 4 times along one of its edges. (b) Unfold the sheet of paper and identify the fixed line. (c) Describe the locus of a point equidistant from the fixed line. (d) State real-life examples of such a locus. Activity 4.1(d) (Work in groups) (a) Fold a sheet of paper / cloth such that two adjacent sides are mapped onto each other. (b) Repeat for the remaining vertices. (c) Describe the locus of points equidistant from two intersecting fixed lines. Activity 4.1(e) (Work in groups) (a) A goat is tied on a tree by a rope of length 1. (b) Describe the area of the grass that will be eaten by the goat. Activity 4.1(f) (Work in groups) Tourists must stand 3 m away from the walls of a rectangular lion enclosure at the zoo. How would you draw an accurate diagram to show where the tourists must stand?

Exercise 4.1

Find the equation of the locus of a variable point P in the following:

(a) P moves such that AP = 4 cm given A(2, 1).

(b) P moves such that BP = CP, given B(2, -1) and C(-4, 3).

(c) P moves such that it is 5 cm away from the line x = -1.

(d) P moves such that it is equidistant from lines x – y = 0 and x + y = 0.

Describe the following loci:

(a) Locus of the centre of a motorcycle wheel while moving along a straight road.

(b) Locus of a goat tethered by a rope which is 2 m long.

(c) Locus of a plane flying at equal distances between two control towers.

4.2 Constructing Loci Involving Points under Given Conditions

Construction involves the use of geometrical instruments.

Activity 4.2(a) (Work in groups)

Which of the following points is inside the circle centred at P(-2, 5) and having a radius of 3 cm?

(a) A(1,9) (b) B(0, 0) (c) C(-6, 8) (d) D(-3, 4)

Activity 4.2(b) (Work in groups) Is it possible to construct a triangle using the line segments below? Explain your answer.

Exercise 4.2

1)Construct the locus of a variable point P such that:

(a) AP = 3 cm, given A(0, 2).

(b) P is 4 cm from the line 2x + 3y = 6.

(c) BP = CP, where Line BC is 8 cm long. (d) P is equidistant from two intersecting lines XY and WZ, intersecting at T.

2) Construct a triangle ABC such that AB = 8 cm, BC = 6 cm, and AC = 10 cm. Find the locus of a point E which is equidistant from:

(a) Points A and B.

(b) Side AC.

(c) Point C.

(d) Sides AB and AC.

3) Two roads X, and Y, are 800 m and 500 m long, respectively and meet at an angle of 300°. If according to the town plan, buildings are to be erected so that they are equidistant from roads X and Y and not less than 100 m from their intersection. Make a scale drawing to show the possible locations of the buildings.

4) Construct the locus of Point X such that AXB = 75° and line AB = 6 cm.

4.3 Constructing Intersecting Loci

Activity 4.3(a) (Work in groups)

(a) Draw two intersecting lines

(b) Describe the locus of points which are equidistant from the 2 lines drawn in (a).

(c) Share your findings with other groups.

Activity 4.3(b) (Work in groups)

(a) Draw a circle with Centre C(-2, 4), and Radius 2 cm.

(b) On that same graph, draw another circle with Centre C(2, 4) and Radius 2 cm.

(c) Where will the two circles intersect? (d) Share your findings with the rest of the class.

Example 4.4

(a) Construct a rectangle ABCD with dimensions AB = 5 cm and AD = 4 cm.

(b) Find the locus of Point P such that P is 3 cm from C and equidistant from Points A and B.

Solution:

The locus is a circle with Centre C and Radius 3 cm.

Exercise 4.3

1)Draw a triangle ABC in which AB = 4 cm, 4B = 105°, ZA = 45°, and produce the sides in both directions. Find, by construction, the points which are equidistant from A and C and also equidistant from BA and BC.

2) Draw a triangle ABC with ZABC = 120°, AB = 7.5 m, and AC 11.4 cm. Draw a perpendicular from C onto AB, meeting AB at D. Find a point P which is equidistant from D and AC.

3) Draw a triangle ABC in which AC = 8 cm, AB = 6 cm, and BAC = 30°. Find Point S which is equidistant from A and C and 8 cm from B.

4) Draw a quadrilateral PQRS in which ZQRS = 45°, QR = 4.5 cm, SR = 6 cm, PS = 7.5 cm and PQ = 10.5 cm.

Find Point T which is equidistant from S and P and in line with R and Q. Find Point U, which is equidistant from Points; R, T, and P.

5) Draw a triangle PQR such that ZPQR = 60°, QR = 9 cm, and PR = 8.5 cm. Find Point O which is equidistant from lines; QR, QP, and PR.

6) Draw a triangle ABC such that BC = 10.6 cm, ZACB = 75° and ZABC= 60°. Find Point D which is equidistant from lines BC and AB and also equidistant from BC and AC. Find Point P which is equidistant from D, touching lines; BC, AB, and AC.

7)Draw a triangle ABC in which AB = 7.2 cm = BC and CA = 5.4 cm.

Find the locus of a point which is equidistant from AB and AC and also equidistant from BA and BC.

8) Construct a triangle ABC in which BC = 5 cm, AB = 4 cm, and the perpendicular distance of A from BC is 2.8 cm. Measure Length AC.

9) Construct an equilateral triangle ABC of side 4 cm. Find, by construction, a point P which is 2 cm from AC and equidistant from BA and BC.

10) A and B are two points which are 3 km apart on a straight road, A being due West of B. A school is 4 km from Road AB and is observed to be on a bearing of N 065°E from A. By construction, find the distance of the school from B

11) Draw an equilateral triangle ABC of Side 7 cm. Hence, find the locus of Point D which is equidistant from all the sides of the triangle.

4.4 Constructing Loci Involving Inequalities Loci are not limited to paths, as discussed before; they can also be represented a regions. A region shows points that are within or greater than a given distance, for example, the region covered by a door while opening. Activity 4.4 (Work in groups) A cow is tethered to the corner of a house. (a) Assuming that the length of the rope is less than the length of the sides of the house, sketch the locus of the area that the cow can reach. (b) If the rope is longer than one of the sides of the house, sketch the locus of the area that the cow can reach. (c) If the rope is longer than both sides of the house, sketch the locus of the area that the cow can reach.

Example 4.6

Given a rectangle ABCD, in which AB = 5 cm and BC = 7 cm, determine the locus of a point which is always at least 4 cm from AB.

Solution:

Let the point be P, P being 4 cm from AB means that the locus of P is a straight line which is parallel to AB and is 4 cm from it. But since, in the question, the distance between P and AB is always at least 4 cm, the required region is that where P is 4 cm or more, from AB.

Exercise 4.4

1) Draw the line y = 2x – 5 on a graph paper. Show the locus of points (x, y) such that;

(a) y + 5 > 2x

(b) 2x-y > 5.

2) With points O(0, 0), A(2, 2) and B(18, 6), form the locus of Point P such that it is nearer to B than to A and not more than 10 units from A.

3) Draw a line AB = 3 cm. Show the region within which Point T lies if AT < 3> TB.

4) PQRS is a square of side 4 cm. D is a point within the square such that PD < 2> QD, and D is nearer to PQ than to PS. Shade the region for all positions of D.

5) On a graph, show, by shading, the unwanted region of all points that are:

(a) more than 6 cm from Point Q(-1, 2)

(b) less than 6 cm from Point R(5, 0)

(c) more than 2 cm from O(0, 0), but less than 6 cm from O(0, 0).

4.5 Finding the Equations of the Loci

A locus has a number of points satisfying a given geometrical condition. Under this subtopic, you will learn how to find the equations of loci.

Activity 4.5 (Work in groups)

(a) Given that a point C is such that its y = 5, determine the locus of this point.

(b) Given that a point V is such that its x and y coordinates are always equal, determine the locus of V.

(c) Given that a point S is such that its x-coordinate is twice its y-coordinate, what is the locus of S? (d) Given two points A(2, 3) and P(x, y) and that AP is 5 units long, what is the locus of P? (e) Display your findings to the rest of the class, including the steps you followed to come up with the loci in (a)-(d) above.

Exercise 4.5

1) Find the locus of a point B such that the sum of its coordinates is 15. Describe the locus.

2) Find the locus of a point T such that the sum of its coordinates is less than 10.

3) Find the locus of a point N such that the sum of the squares of its coordinates is 25.

4) Find the locus of a point G such that the distance of G from the x-axis is equal to 10 times its distance from the y-axis.

ICT Activity / Project

(a) One of the ancient weapons that was used to hunt wild animals was a bow and arrow. Use the environment around you to construct a bow and arrow instrument.

(b) Make a PowerPoint presentation about the use of the weapon in Uganda.

Revision Questions:

1) The diagram shown is of a garden measuring 10 m by 10 m. A dog is tied up in this garden. 10 m using a chain of length 3 m. Draw the garden and the possible areas the dog can cover if it is tied up in any of the four corners. Label the region that it cannot access as R.

2) Construct a quadrilateral ABCD in which AB = 8 cm, BC = 9 cm, CD = 10 cm, DA = 11 cm, and angle DAB = 102°. Use a ruler and a pair of compasses to find the point P which is at the same distance from BC and CD and is 7 cm from A. Find the length of PD in millimetres, correct to the nearest millimetre.

3) Construct the locus to show the area available to a cow tethered by a rope of length 3 m to a horizontal rail of length 4 m. Find the area of this locus.

4) Juma is planning to put a fountain in his garden. His house is situated along the line CD and there is a tree at Point E. He wants the fountain to be at least 3 m away from his house and at least 1.5 m away from the tree. Shade the locus of the points where he can have the fountain.