Properties of Circle

Geometrical properties of Angles ( revision)


1. Two angles are complementary angles if their sum equal to 90 degrees.
2. Two angles are supplementary if their sum is equal to 180 degrees.
3. The sum of adjacent angles on a STRAIGHT line is equal to 180 degrees.
4. Vertically opposite angles are equal.
5. The sum of angles at a point is equal to 360 degrees
Corresponding angles are equal!
6. Alternate angles are equal while interior angles are supplementary

As a student myself too, i feel that the corresponding angles and knowing how to see and spot an isosceles triangle in solving circle questions is very important


Some questions


1. Are equal chords equal distance from the center?


The reason why students mix this up is because the chords do not seem to be of equal length. They feel that they ate not equal.


Ans: Yes, equal chords are of equal distance from the center of the circle.


2. Any angle subtended at the circumference is half the angle subtended at the center.


Some students feel that way, they see a angle at the circumference and say it is half the one at the center.


Ans: However, this is wrong. The circumference subtended at the circumference is only half of the angle subtended at the center when they are separated by the SAME CHORD.


3. A cyclic quadrilateral is any four sided shape drawn inside a circle.


When students do not know their facts well, they tend to make their own rules. This is when they do not know how to solve a question and resort to making their own rules.


Ans : A cyclic quadrilateral is a quadrilateral drawn inside a circle so that all four of its vertices lie on the CIRCUMFERENCE of the circle.

4. Similar to a trapezium and parallelogram. In a cyclic quadrilateral, the angles at the opposite, but same side of the quadrilateral are supplementary.

Students mix up the cyclic quadrilateral with the parallelogram and trapezium. Whereas in a cyclic quadrilateral, the opposite sides are not necessarily parallel, hence the angles at opposite but same sides of the quadrilateral are NOT supplementary. Instead, it is the opposite angles in a cyclic quadrilateral which add up to 180 degrees, hence supplementary.

From my last previous test, i feel that one has to take special attention to rule 2. I lost 6 marks because i made a observation of angles in the same segment

It should be angle in the center is twice angle at circumference.

Quadratic equations and Quadratic graphs

Roots of a Quadratic Equation
Definition of Roots of a Quadratic Equation

• The solutions of a quadratic equation are called the roots of a quadratic equation.

Example on Roots of a Quadratic Equation

• The solutions or the roots of the quadratic equation x² - x - 2 are 2 and - 1.

1. We find the coordinates of X and Y on the graph by naming one of them = 0

2. When the graph of y = ax ^2 + bx + c touches the x-axis at one point, we cant say that the roots of the equation is repeated.

3. When the graph of y = ax ^2 + bx + c does not touche the x-axis at one point, we cant say that the equation has no real roots.

Also, students please remember to write down the specifics for one's own graph like writing 1cm= to represent 2 units!

Methods to solve quadratic equations

1 . Solving the quadratic equations by factorization
Students must aim to write the quadratic equation in the general form and put all terms on one side and 0 on the other side before factorizing.

2. Solving the quadratic equations by completing the square

For example. ( x-3 ) ^2 = 25
x-3 = plus, minus 25
x-3 = plus minus 5
hence , x-3= 5 or x-3 = -5

Tips for quadratic equations solving.

1. Always express the quadratic equation in the general form
2. When applying to some questions , we might have some rejected answers like negative answers when they asking for length.

When solving, we can recall the rules we learnt earlier.

Gradient of a Straight Line

1. The gradient of a straight line is the measure of its steepness of the slope
2. The gradient is the ratio of the vertical distance.

Remember ! gradient = Rise/ Run

3. To find the gradient m , of the line passing through 2 points, A ( x1, x2 ) and B ( X2, Y2 )
Use the formula Gradient m , = y2-y1 / x2-x1.

Y= mx + c
m= gradient while
c = y intercept

Line of intercept = The middle, commonly the minimum or the maximum point !
Also another tip: Always remember to draw the graph enlarged so that the points is clearly shown

Linear Inequalities

An inequality has the same characteristics as an equation except that in the place of the equal sign , there is an inequality sign.

Inequality signs !

1. < '' is less than'' 2. > '' is greater than''

A tip : One can add or subtract the same number, positive or negative on both sides of the eauality without changing the inequality sign !

This is a mistake a lot students make like me.


* NOTE THIS !

1. When one multiply or divide both sides of the inequality by the same positive number, the inequality remains unchanged

2. HOWEVER! When one multiply or divide both sides of the inequality by the same NEGATIVE number, the inequality sign is reversed!


An example of a question involving number line .

Question : solve the inequalities 2x + 5 > 11 and 5x-7< 18

Note than students should solve both separately then combine on the number line to reduce mistakes.
Tip : stretching the individual solutions on a number line will help one to visualize the solution for simultaneous linear equations.

Standard Form

A standard form is a convenient way to write a very large or very small numbers .
It is quite similar to indices whereby the number has a base and index.

A important observation from me . Students often forget whether to move the decimal place to the left or to the right.


So, i came up with steps to write a number in standard form

1. Move the decimal point until one gets a number between 1 to 10
2. Count the number of places one moves the decimal point.
The exponent is POSITIVE if one moves the point to the left while the exponent is NEGATIVE if one moves the point to the right! Take note!

Also, students must remember that a number expressed in standard form is a product of a number between 1 and 10, therefore , it is also a product of 10. I have 3 examples.

1 . 3862.9 = 3.8629 x 10^3 . Here , we move the decimal 3 places to the left.
2. 978000= 9.78 x 10^5 . Here , we move the decimal 5 places to the left.
3. 0.000568 = 5.68 x 10^-4 . Here , we move the decimal 4 places to the right.

Below, i have started the first steps for a few questions . Why don't you try to solve it?
Addition :

3.2 x 10^3 + 2.5 x 10^2
= 32 x 10^2 + 2.5 x 10 ^2 ... Here we extract the lowest power of 10 and change them to the same power

Subtraction :

5.68 x 10^-3 - 4.3 x 10^-4
= 56.8x 10^-4 - 4.3 x 10^-4
Here we extract the lowest power of 10 . also, we do that by moving the decimal to the left .

Multiplication :

(2 x 10^-2) x ( 5.6x 10^5 )
= (2 x 5.6) x ( 10 ^-2 x 10 ^5 ) ... remember rules of indices? it works here ! Collect all the powers of 10 together and the rest into another group .

Then ...

= 11.2 x 10^3
= 1.2 x 10^4 ... Here students must remember not to stop at 11.2 x 10^3. this is because it is a answer in standard form

Remember standard form = 1.0-9.99 x 10 ^something.



Questions


1. How do we use a calculator to evaluate a number in standard form?


The main problem that students face when solving standard form questions are when the do not know how to type into their calculator, hence causing them to know how to solve the question but not knowing how to use the calculator to derive at the correct answer.


Ans: We use the EXP key on the calculator to evaluate a number in standard form.

Topic 1 : Indices

Welcome Ms wong, this is a blog i created from scratch explaining maths i have learned over this one year

Over the year, now we are approaching the end of Years exams!

Term 1 - We learnt and linear inequalities with indies from Ms Wong

Term 2- We learnt quadratic equations and quadratic graphs from Ms tan

Term 3- We learnt about similar figues, solids, congruent and similar triangles as well as Properties of Circles from Ms Wong