Preparation For Ujian Cermin Diri 2

Loci In Two Dimension
EXPRESS NOTES

1. The locus of a point is the path traced out by the moving point under given geometrical condition (or conditions). Alternatively, the locus is the set of all those points which satisfy the given geometrical condition (or conditions).

2. TYPES OF LOCUS
The locus of a point, which is equidistant from two fixed points, is the perpendicular bisector of the straight line joining the two fixed points.
The locus of a point, which is equidistant from two intersecting straight lines, consists of a pair of straight lines which bisect the angles between the two given lines.

EXERCISE

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

(i) The locus of points at a distance 2 cm from a fixed line.

(ii) The locus of points (in a plane) at a constant distance 2 cm from a fixed point in the plane.

(iii) The locus of points (in space) at a constant distance 2 cm from a fixed point.

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

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

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

(vii) The locus of center of a circle of varying radius and touching two arms of ABC.

(viii) The locus of center of a circle of varying radius and touching a fixed circle, center O, at a fixed point A on it.

(ix) The locus of center of a circle of radius 2 cm and touching a fixed circle of radius 3 cm with center O.

2. A, B and C are three collinear points.

(i) Construct the locus of point equidistant from A and B.

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

(iii) Is it possible to locate a point equidistant from A, B and C.

3. Take PQ = 6 cm. Construct

(i) the locus of points equidistant from P and Q.

(ii) the locus of points 5 cm from P.Mark the two points which lie on both loci and measure the distance between them.

4. In a triangle ABC, the right bisectors of AB and BC meet at P.

(i) Assign the special name to the point P.

(ii) Prove that PA = PB = PC.

(iii) If ABC = 90°, find the exact location of the point P with respect to the side AC.

5. Construct a triangle PQR with PQ = 5 cm, QR = 4 cm and RP = 3·6 cm. Find by construction a point P which is equidistant from the three vertices P, Q and R.

6. Construct a triangle ABC in which BC = 3·8 cm, CA = 4 cm and AB = 5·1 cm. Find by construction a point P which is equidistant from BC and AB, and also equidistant from B and C.

7. Construct a triangle ABC, in which AC = BC = 5 cm and AB = 6 cm. Locate the points equidistant from AB, AC and at a distance 2 cm from BC. Measure the distance between the two points obtained.

8. Construct a triangle ABC, in which AC = BC = 5 cm and AB = 6 cm. Locate the points equidistant from B, C and at a distance 2 cm from A. Measure the distance between the two points obtained.

9. Construct ABC = 75°. Mark a point P equidistant from AB and BC such that its distance from another line DE is 2·3 cm.

10. Draw two intersecting straight lines to include an angle of 135°. Also locate points which are equidistant from these lines and also 1·8 cm away from their point of intersection. How many such points exist?

11. AB and CD are two intersecting st. lines. Locate points which are at distances 2·5 cm and 1·8 cm from AB and CD respectively. How many such points are there?

12. 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.

13. 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 P on AD such that PB = PC. Measure the length AP.

14. ABCD is a rhombus with side 4 cm and ABC = 120°.

Construct the locus of points inside the rhombus

(i) equidistant from A and C.

(ii) equidistant from B and D.

(iii) equidistant from A and B.

15. Draw a line segment AB of length 6 cm, M is mid point of AB. Construct:

(i) the locus of points 3 cm from AB

(ii) the locus of points 5 cm from M.

Mark two points P and Q on the same side of AB satisfying the above loci. Measure the distance between P and Q.

16. AB is a fixed line. Draw and describe the locus of the center of a circle of radius 2·5 cm and touching the line AB.

17. If the diagonals of a quadrilateral bisect each other, prove that the quadrilateral is a rhombus.

18. 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.

19. 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.

20. Use ruler and compasses only for the following question:Construct triangle BCP, where CB = 5 cm, BP = 4 cm, PBC = 45°.

Complete the rectangle ABCD such that

(i) P is equidistant from AB and BC, and

(ii) P is equidistant from C and D.Measure and write down the length of AB.

Answers

1. (i) A pair of straight lines parallel to the given line.

(ii) A circle with fixed point as center and radius 2 cm.

(iii) A sphere with fixed point as center and radius 2 cm.

(iv) Perpendicular bisector of the line segment joining given points.

(v) Diagonal AC of the rhombus ABCD.

(vi) Diagonal BD of the rhombus ABCD.

vii) The bisector of ABC.

(viii) The straight line passing through O and A.

(ix) Concentric circle of radius 1 cm if circles touch internally; and concentric circle of radius 5 cm if circles touch externally.

2. (iii) No

3. 8 cm

4. (i) circumcenter

(iii) Mid-point of AC

7. 4·1 cm (app.)

8. 3·4 cm (app.)

10. Four

11. Four

12. (i) 65°.

13. 78°; 1·2 cm

14. (i) Diagonal BD (ii) diagonal AC (iii) right bisector of AB

15. (i) Pair of straight lines parallel to AB and at distance 3 units on either side of AB (ii) Circle with center M and radius 5 cm; 8 cm

16. Pair of straight lines parallel to AB and at distance 2·5 cm on either side of AB18. 5·25 cm approximately

19. (i) The diameter of the circle right bisecting AC (ii) The segment of the circle bisecting BAC.

20. 5·5 cm (app.)

Peeling Potatoes

Loci In two Dimension
Loci 1


Loci 2

Cartesian Coordinate
Revision Exercise

1 . Diagram 1 shows a Cartesian plane.

Diagram 1
Based on the Cartesian plane in diagram 1;

a. state the coordinates of A, B and C

b. find the distance from A to the y-axis

c. find the midpoint of AC

d. if the midpoint of BD is (-2,0), what is the coordinate of D

2. In diagram 2, PQ = QR.

Diagram 2
Find

a) the length of PQ.

b) the coordinates of point R.

c) the midpoint between points Q and R.

3. In diagram 3, points A, B and D are three of the vertices of a rectangle.

Diagram 3

State the coordinates of the fourth vertex of the rectangle.

Coordinate System



Distance Between Two Points



Finding The Midpoint

Math Challenge 1
27th June - 3rd July
Three challenges to set you thinking: How many can you solve...........?

No. 1: "A Perfect Match"

In the diagram below, 11 matches make 3 squares:
Your challenge is to move 3 matches to show 2 squares.

No. 2: "Nenek’s age"

Mamat asked his nenek how old she was. Rather than giving him a straight answer, she replied: "I have 6 children, and there are 4 years between each one and the next. I had my first child (Pak Long Husin) when I was 19. Now the youngest one (Mak Su Jamilah) is 19 herself. That's all I'm telling you!". How old is Mamat's nenek?

No. 3: "Peas Galore"

At a school fete people were asked to guess how many peas there were in a jar. No one guessed correctly, but the nearest guesses were 163, 169, 178 and 182. One of the numbers was one out, one was three out, one was ten out and the other sixteen out.
How many peas were there in the jar?

e-mail your answers to your teacher before 4th July 2009

Baking By Numbers
Not all people are chefs, but we are all eaters. Most of us need to learn how to follow a recipe at some point. To create dishes with good flavor, consistency, and texture, the various ingredients must have a kind of relationship to one another. For instance, to make cookies that both look and taste like cookies, you need to make sure you use the right amount of each ingredient. Add too much flour and your cookies will be solid as rocks. Add too much salt and they'll taste terrible.

Ratios: Relationships between quantities
That ingredients have relationships to each other in a recipe is an important concept in cooking. It's also an important math concept. In math, this relationship between 2 quantities is called a ratio. If a recipe calls for 1 egg and 2 cups of flour, the relationship of eggs to cups of flour is 1 to 2. In mathematical language, that relationship can be written in two ways:
1/2 or 1:2
Both of these express the ratio of eggs to cups of flour: 1 to 2. If you mistakenly alter that ratio, the results may not be edible.

Working with proportion
All recipes are written to serve a certain number of people or yield a certain amount of food. You might come across a cookie recipe that makes 2 dozen cookies, for example. What if you only want 1 dozen cookies? What if you want 4 dozen cookies? Understanding how to increase or decrease the yield without spoiling the ratio of ingredients is a valuable skill for any cook.

Let's say you have a mouth-watering cookie recipe:

1 cup flour

1/2 tsp. baking soda

1/2 tsp. salt

1/2 cup butter

1/3 cup brown sugar

1/3 cup sugar1 egg

1/2 tsp. vanilla1 cup chocolate chips

This recipe will yield 3 dozen cookies. If you want to make 9 dozen cookies, you'll have to increase the amount of each ingredient listed in the recipe. You'll also need to make sure that the relationship between the ingredients stays the same. To do this, you'll need to understand proportion. A proportion exists when you have 2 equal ratios, such as 2:4 and 4:8. Two unequal ratios, such as 3:16 and 1:3, don't result in a proportion. The ratios must be equal.

Going back to the cookie recipe, how will you calculate how much more of each ingredient you'll need if you want to make 9 dozen cookies instead of 3 dozen? How many cups of flour will you need? How many eggs? You'll need to set up a proportion to make sure you get the ratios right.
Start by figuring out how much flour you will need if you want to make 9 dozen cookies. When you're done, you can calculate the other ingredients. You'll set up the proportion like this:

(1 cup flour /x cups flour ) x 3 dozen/9 dozen

You would read this proportion as "1 cup of flour is to 3 dozen as X cups of flour is to 9 dozen." To figure out what x is (or how many cups of flour you'll need in the new recipe), you'll multiply the numbers like this:

x times 3 = 1 times 9,

Therefore 3x = 9

Now all you have to do is find out the value of x. To do that, divide both sides of the equation by 3. The result is X = 3. To extend the recipe to make 9 dozen cookies, you will need 3 cups of flour. What if you had to make 12 dozen cookies? Four dozen? Seven-and-a-half dozen? You'd set up the proportion just as you did above, regardless of how much you wanted to increase the recipe.
What if your recipe has metric measurements? Find out more about the metric system in

"Meters and Liters: Converting to the Metric System of Measurements."

Source:http://www.learner.org/interactives/dailymath/cooking.html

All About Patterns
Mathematics has been called 'science of patterns'. Recognizing and describing patterns and using patterns are important mathematical skills.

A Bee Tree
Although a female honeybee has two parents, a male honeybee's ancestors reveals an interesting pattern of numbers.

1 bee

1 parent

2 grandparent

3 great-grandparent

5 greatt-great-grandparent

8 great-graet-great-grandparent

The numbers of bees in the generation: 1,1,2,3,5,8.......form a famous list of numbers knowns as the Fibonacci sequence. This number sequence has fascinated peoples for centuries because it appears so often in nature.

You can find numbers from the Fibonacci sequence in the patterns of leaves on plants, in the arrangement of scales pineapples and in the spirals of nautilus shells

The surface of a honeycomb is made up of a pattern of hexagons that fit together with no overlaps

About Fibonacci
Leonardo Fibonacci was born in Pisa, Italy, around 1175. His father was Guilielmo Bonacci, a secretary of the Republic of Pisa.His father was also a customs officer for the North African city of Bugia. Some time after 1192. Bonacci brought his son with him to Bugia.Guilielmo wanted for Leonardo to become a merchant and so arranged for his instruction in calculational techniques, escpecially those involving the Hindu - Arabic numerals which had not yet been introduced into Europe.Since Fibonacci was the son of a merchant, he was able go travel freely all over the Byzantine Empire. Merchants at the time were immuned, so they were allowed to move about freely. This allowed him to visit many of the area's centers of trade. While he was there, he was able to learn both the mathematics of the scholars and the calculating schemes in popular use, at the time.

MATHS IN HOME DECORATING

What does math have to do with home decorating? Most home decorators need to work within a budget. But in order to figure out what you'll spend, you first have to know what you need. How will you know how many rolls of wallpaper to buy if you don't calculate how much wall space you have to cover? Understanding some basic geometry can help you stick to your budget.

The word geometry literally means "to measure the Earth." Geometry is the branch of math that is concerned with studying area, distance, volume, and other properties of shapes and lines. If you need to know the distance between two points, the volume of water in a pool, the angle of a tennis serve, or how much wallpaper it will take to cover a wall, geometry holds the answers.

Figuring area: Squares and rectangles

Imagine you're planning to buy new carpeting for your home. You're going to put down carpeting in the living room, bedroom, and hallway, but not in the bathroom. You could try to guess at how much carpet you might need to cover these rooms, but you're better off figuring out exactly what you need. To determine how much carpet you'll need, you'll use this simple formula:

Area = Length x Width

This formula is used to determine the area of a rectangle or square. In the floor plan below, all of the floor space (as well as the walls and ceilings) is made up of squares or rectangles, so this formula will work for figuring the area you need to carpet.

Start by figuring the total area of the floor plan. When you're done, you can deduct the area of the bathroom, since you don't want to carpet that room. To figure out the total area of the floor plan, you'll need to know the total length and width. The total length of the floor plan shown above is 12 feet plus 10 feet, or 22 feet. The total width is 7 feet plus 5 feet, or 12 feet. Plug these numbers into your equation to get the total area of the floor plan:

A = 22feet x 12 feet = 264 sq feet

The total area of your floor plan is 264 square feet. Now you need to figure out the area of the bathroom so you can deduct it from the total area. The bathroom is 7 feet long and 5 feet wide, so it has an area of 35 square feet. Deducting the area of the bathroom from the total area (264 minus 35) leaves you with 229 square feet to carpet.

Figuring Area : Circles

Calculating how much carpet you'll need is a fairly simple task if your home has only square or rectangular rooms. But what if you have a circular alcove at the end of one room? How do you figure the area of a circle? Use this formula:

A = pi x r²

In English, this formula means "area equals pi times the radius squared." A circle's radius is one half of its diameter, or one half of what you get if you measure all the way across its widest part. "Squaring" something means you multiply it by itself. Pi is a number that roughly equals 3.14159.

If your living room has a semi-circular alcove as shown in the floor plan above, you'll need to use this additional equation to figure out its area. To figure the radius of your alcove, the number you'll need to plug into the equation, you'll divide its diameter in half. Its diameter is the same as the width of the living room: 12 feet. Half of that is its radius: 6 feet.
Let's plug in the numbers:

A = 3.14159 x (6 feet x 6 feet)

A = 113 square feet (rounded to the closest square foot)

If your alcove were a complete circle, it would have an area of 113 square feet. Because it's a half circle, its area is half of that, or 56.5 square feet. Adding 56.5 square feet to the rest of your floor plan's area of 229 square feet gives you the total area you want to carpet: 285.5 square feet. Using geometry, you can buy exactly the amount of carpet you need.

http://www.learner.org/interactives/dailymath/decorating.html

How To Study Mathematics
You can learn mathematics better if you do certain things. A lot of people never learn these basic approaches that can open up the world of math.

Steps
1. Learn the vocabulary!
Every area of Math has its own vocabulary. You should memorize every definition, word for word. (Don't leave anything out.) You should be able to recite them. If you can recite them, and if you can write them all out, then you know them. If you can't do that, then you don't know them. This might sound like a lot of work, but it is nothing compared to the memorizing in history or geography.

2. Try to work the problems and do the exercises, at least some of them.

If there is a sample problem in your book, or an example problem, work through it yourself,

and use the sample in your book to guide you.

3. Identify the error.

If you make a mistake in a problem, or in a proof, or take a wrong turn, figure out why! Figure

out what it was, or what it was that you were thinking, that led you down the wrong path

4. Avoid the risk of frustration.

Don't go on in your book until you have learned the material where you are now. The later stuff builds on the earlier stuff. A math book is like a novel, it doesn't make sense unless you start at the beginning.

5. Write down your question, if you don't understand a specific concept.

By identifying what you need to learn, you'll speed up the learning process.

6. Be able to explain it to somebody else.

This is a sure way to find out what you don't understand.

WARNING!!!!!!!!

Don't be afraid to ask for help from someone who has been doing math for a long time or is further along.

• Don't be too hard on yourself. Realize that many have struggled with the same areas you are learning. Some people just take longer to understand math. Eventually, with enough, perserverance, you can succeed in math.

CARTESIAN COORDINATES

The Cartesian coordinate system was developed by the mathematician Descartes during an illness. As he lay in bed sick, he saw a fly buzzing around on the ceiling, which was made of square tiles. As he watched he realized that he could describe the position of the fly by the ceiling tile he was on. After this experience he developed the coordinate plane to make it easier to describe the position of objects.

Cartesian Coordinate System consists of two axes, X and Y, which intersect each other at a point called `origin`, and is used to define the position of any point by using ordered pairs. In two-dimensional coordinate system, the reference of a point is given by using two coordinates, X and Y.

Who Uses Coordinates?

The system of coordinates that Descartes invented is used in many modern applications. For example, on any map the location of a country or a city is usually given as a set of coordinates. The location of a ship at sea is determined by longitude and latitude, which is an application of the coordinate system to the curved surface of Earth. Computer graphic artists create figures and computer animation by referencing coordinates on the screen.