MTH4: CONSTRUCTION AND LOCI PLUS VARIATIONS

This unit is about construction and loci where a locus of a point is the path it describes and variations.

CONSTRUCTION AND LOCI

A locus of a point is the path it takes as it moves following the given conditions. (Law)

  1. A point P moving such that its distance from a given point is constant. e.g. let the distance of P from O be 2m.

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There fore the locus of a point P a distance x from a given point O. Is a circle with O as the centre and x as radius.

2. A point P moving such that its distance from a given line is constant e.g a point P is 20 m from the line below.

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Therefore the locus of a point at a given distance from a given line would be a pair of straight lines both parallel to AB and at a given distance from AB.


3. A point moving such that its distance from two fixed points AB is equal.

4. A point moving such that its distance from two given intersecting straight line is equal.


The locus of a point equal distant to two intersecting lines is a bisection of the angle between the two lines.
Examples.

1. Draw a line XY of a length 10 cm. Construct the locus of a point which is equidistant from x and y.
2. Draw two lines AB and AC of length 8 cm, where , BAC = 600. Construct the locus of appoint which is equidistant from the lines AB and AC.
3. Draw a circle , centre O of radius 5 cm and a radius OA. Construct the locus of appoint P. which moves so that OAP = 900
4. Draw a line AB of length 10 cm and construct the circle with diameter AB. Indicate the locus of a point P which moves to that APB = 900
Exercise
1. Inspector Clouseau has put a radio transmitter on a suspects car, which is parked some where In Gayaza. From the strength of the signals received at Points R and P, Clouseau knows that the car is
a) not more than 10 km from R and
b) not more than 20 km from R
Make a scale drawing (1 cm=10 km) and show the possible positions of the car .
2. A goat B tied to one corner on a grazing filed. The diagram below show the plan view.
Sketch a plan showing where the goat can graze if the rope is 4 cm long.

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ii) By construction determine the centre and radius of the circle
iii) Calculate the area of the minor segment cut off by the chord PQ

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A VIDEO BELOW ABOUT CONSTRUCTION AND LOCI

VARIATIONS

Proportions

  1. Direct Proportion.

Given that P is directly proportional to Q , we can also say; p varies as q and write;

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JOINT PROPORTION
A quality varies with more than one quality.
Examples
Given qualities Q, P and V.
Joint direct variation.
Q and also p varies directly with v then i

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2. Joint inverse variation.
If P varies inversely with q and varies inversely with v;

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Examples
1. The power P of an electric appliance varies directly as the square of the voltage , v and inversely as the resistance Express this as an equation.

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Z varies directly as y and inversely as t . When Z =6 , y=4 nad t=5 find z when y=24 and t=20.

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3. The weight of a cuboid varies directly as the height and the area of the base. If the height is increased by 20% and the area of the base by 10%. Find the percentage change in weigh

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A VIDEO BELOW ABOUT DIRECT AND INVERSE VARIATION

Part Variation
Under this variation, one quality may be expressed as a sum of
two different qualities. At times one being a constant.
Examples
Given that P is partly constant and partly varies as Q , then
p=a+bq,where a and b are constants.
If t partly varies with and partly varies v then.
t=ks+hv
Where k and h are constant.
P is partly constant and partly varies directly as the square of t
Given that p= 18 when t = 2 and P= 82 when t=6. Find P when t = 5

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The monthly cost in shillings of running a car is partly constant and partly varies directly as the mileage in km. Given that the
Cost was sh 6400 for 500 km in August and the cost was 7200 for 700 km in September,

Find the mile age in October if the cost was sh 8400.

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5. When a bus is running along horizontal ground at a speed of 5km/hr. The distance d in meters in which the bus can be stopped varies partly as the speed and partly as the square of the speed.
If it’s running at 20km /hr the distance required to stop the bus is 75m. How ever if the bus is travelling at 50km/hr ∝ distance 135m is needed.
Find the distance required to stop a bus travelling at 80km/h speed of the bus – S, distance required – d

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= 0.284 km

Exercise
2. The area of a circle varies directly as the square of its radius.
a) Find the percentage change in the area of the radius is increased by 50%

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c) If the area decreased by 19% what is the percentage change in the radius.

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3. The number of beats per minute of pendulum varies inversely as the square root of its length if the pendulum is 8/cm long it makes 24 beats per minute. Calculate how many beats per minute a pendulum of 25cm will make

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=43.2 beats per min

 

4. The pressure of a given mass of gas at constant temperature is inversely proportional to its vol. If the pressure is increased by 30% . Find the percentage chance in vol.

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5. The heat capacity c in calories developed in a copper wire by an electric current varies jointly as time (t) seconds and as the square of the voltage v volts and inversely as the resistance R in ohmns for a wire the R for 4 ohmns, the heat developed in 54 seconds with voltage of 45 volts in 60seconds calories. Find how much heat is developed in 75 seconds in a wire with 90 volts.

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6. The greatest weight in which can be carried by a beam varies directly as the cube of its depth d and inversely as its length. If w increases by 20% and d increases by 30%. Determine the percentage change of length of beam.

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Exercise (Part Variation)
1. The time t, taken to harvest coffee beans on a rectangular field and party as the number of coffee beans on the fields.
d represents coffee beans on rectangular field.
n represents the number of coffee beans on the fields.

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ASSIGNMENT : CONSTRUCTION AND LOCI PLUS VARIATIONS ASSIGNMENT MARKS : 50  DURATION : 3 hours

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