# Distance, Mid-point and Circles

## Day 1: Distance and Mid-point# Table of Contents Day 1: Distance and Mid-pointtoc Day 2: Equation of Circles Homework ACE Project: GC Design

You have already learnt to calculate the equation of a line given 2 points - by finding gradient and y-intercept.

Given the coordinates of 2 points, we can also calculate it's length, as well as the coordinate of the mid-point.

NOTE: If the videos do not load or are slow, you can click on the URLs to view them directly from YouTube.

[20 Mins] Concepts and Formulas:
(1) View this video for the Distance Formula

[URL: http://www.youtube.com/watch?v=9GGf2be54AI ]

and this for the Mid-point Formula:

[URL: http://www.youtube.com/watch?v=jSAKvKsD--A ]

(2) The powerpoint slides show a summary of the formulas, as well as examples of how to use them. You can use the slides as a reference for your practice. Click on the slides itself to navigate.

[30 mins] Practice:

Complete these questions without plotting the points on graph paper. Give your answers correct to 3 s.f if necessary.

Q1) Find the distance and mid-point of the following 2 points.
(a) A (2, 3) and B (9, 7)
(b) C (–1, 4) and D (8, –3)

(c) E (–8, 2) and F (–8, –17)

Q2) Find the perimeter of ∆ABC, whose vertices are A (–4, –2), B (8, –2) and C (2, 6).

Q3) Given that M (p, 7) is the mid-point of the line joining the points A (–3, 1) and B (11, q). Find the values of p and q.

Q4) Three of the vertices of a parallelogram ABCD are A (–3, 1), B (4, 9) and C (11, 3). Find
(a) the mid-point of the diagonal AC.
(b) the fourth vertex D.
(c) the length of the diagonal AC.
(d) the perimeter of ABCD.

Q5) If the distance between the points A (k, 0) and B (0, k) is 10. Find the possible values of k.

[10 mins] Reflection

Complete your reflection in your OWN page: http://20102p1math.wiki.hci.edu.sg/ or http://20102p2math.wiki.hci.edu.sg/

## Day 2: Equation of Circles

You have learnt the equation of linear and quadratic graphs. Now let's move on to the equation of a circle. Knowing this will give you an alternative in GC design.

[20 mins] Concept and Formulas

As you have learnt,
the general equation of a straight line is y = mx + c, where m is the gradient and c is the y-intercept.
the general equation of a quadratic curve is y = ax^2 + bx + c, where a, b and c are constants. a tells you whether it is a smiling face or a sad face and also affects the width of the curve. b will shift the curve either left or right and c will shift the curve either up or down.

The general equation of a Circle is
, where a and b are the coordinates of the centre and r is the radius.

(1) This video shows how to derive the equation of a circle from the distance formula:

[URL: http://www.youtube.com/watch?v=HjN9TTRrQiA ]

(2) Download this excel applet and move the scrollbars to see how the variables affect the attributes of the Circle.

(3) The powerpoint slides will summarise the equation and show you examples of how it works. Click on the slides to navigate.

[30 mins] Practice

Now that you have learnt the equation of circles, we shall use the olympic rings as a practice. Using the equation of circles, re-create the design of the 5 olympic rings in your GC. (Do not use Geogebra else it defeats the purpose of you learning how to form the equation).

To start, plan your drawing on a piece of paper. You need 5 circles, means you need to come up with 5 equations. Decide on the coordinates of the centre of each circle and also the radius of your circles.

Using the equation of circle, generate the equation for all 5 circles.

To enter into the GC, you will notice that you cannot just key in the whole equation. Every equation in the GC must begin with "Y=". Hence, you need to change the subject of the circle equation, such that you will have "Y=....."

Remember that when you take square-root, you need to add a +/- sign. Hence, for each circle equation you need to key in 2 separate equations, one for the positive root and one for the negative root.

An example of how to change the subject and key into the GC:

Hence, you will require 10 equations to draw the 5 circles.

Once you have completed, do a screen capture of your design and upload it onto the student submission page, according to your index numbers.

[10 mins] Reflection

Complete your reflection in your OWN page: http://20102p1math.wiki.hci.edu.sg/ or http://20102p2math.wiki.hci.edu.sg/

## Homework

For your homework, you have to complete these 2 designs.

Graphic 1

Graphic 2

Submit your work on either http://20102p1math.wiki.hci.edu.sg/ or http://20102p2math.wiki.hci.edu.sg/ according to your index numbers. You will need to capture the window settings as well as the equations you used.

If you have time and wish to explore more, you may wish to attempt Graphic 3. You can google to find out how to draw the "waves". The vertical line can be graphed by using the equation of a very steep line (since the GC does not allow you to start an equation with "X=".)

Graphic 3

## ACE Project: GC Design

For term 3, those who have selected Math ACE can opt to submit a GC Design. Since VJC is organising an external competition, we shall adopt their theme for 2010: GC and Olympism. Design a graphic with the YOG or the olympic spirit in mind! There are attractive prizes to be won in the external competition.

Visit the website to learn more and have a look at past winner's work:

http://www.vjc.moe.edu.sg/academics/dept/mathematics/design/
http://www.vjc.moe.edu.sg/academics/dept/mathematics/design/results2009.html

You may embark on your project using the virtual GC. Should your work be judged as substantial for the external competition, you will be asked to re-create your design using the handheld GC (as per competition rules).

This project is strictly INDIVIDUAL WORK.

If you wish to take part in the external competition + ACE, submission deadline is Term 3 Week 1 Friday 2nd July.

If you are submitting as a normal ACE project, submission deadline will be somewhere towards the end of term 3 - date will be given later.

Pls adhere to the submission guidelines stated on the Graphing Calculator Design Page.

Use the June Holidays to come up with a good design!