GSP_S2

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1

Project Maths Development Team 2009

INTRODUCTION

When you open a sketchpad document you will get a similar screen to the one in Figure 1.

Figure 1

When starting Sketchpad it is a good idea to familiarise yourself with the contents of the

Drop Down Menus and the Toolbox.

The Drop Down Windows.

These menus contains many commands. Familiarity with what commands are available and

where they are locatedis quite useful. The black writing indicates that those commands are

available, while the gray writing indicate that those commandsare not available (i.e. youneed to select some geometric objects on which to apply these commands before they

become available).We will now examine these menus.

Drop Down Menus

Toolbox

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File Menu

The File menuis the standard file menu for all windows

applications. You can Open, Save and Print Sketchpaddocuments using this menu.

EditMenu

The Edit menuhas the standardcommands such as Cut, Copy

and Pastebut it also has some

very important extra commands

which are highlighted in the text

boxes.

Setting Preferences

Select Edit and Preferences. The dialog boxon the

right will open.From this you can select the

preferences you want.

Enables you to animate

selected objects on a sketch.

Very important command,

which should be accessed atthe beginning of a sketch. Itenables you to set your

preferences (Number ofdecimal places, Colour and

Font size).

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DisplayMenu

The Displaymenuas the name

suggests and as can be seen from the

menu on the right allows you to

change colour, thickness of lines and a

host of other display options.

ConstructMenu

The Construct menuallows one to construct various elements in

geometry such as mid-points, intersections, lines, rays, parallel

lines, perpendicular lines, angular bisectors, circles and arcs.

Transform Menu

The Transformmenuallows one to construct the images of

objects under translations, symmetries and rotations.

Enables you to show,

hide or label objects .

Enables you to

control the speed of

animating objects.

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Measure Menu

The Measure menuallows one to measure the properties of

objects. As we can see from the menu some of the things we

can measure are length, perimeter, angles, radii, ratio, slopes

and we can find equation. NB when you click on Calculate a

calculator appears which we can use to perform operations.

GraphMenu

The Graphmenuallows one to havea Cartesian Plane and

this menu enables you to draw graphs and plot functions.

Calculator

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Window and Help Menus

These are standard in all windows bases programs.

The Toolbox

The toolbox,as seen in the diagram below,is the column on the left of the screen, contains

several tools that are frequently used in geometric constructions in GSP (selection, dragging,

ruler, compass, labels, etc.). An arrow pointing to the right, as before, indicates that there are

more commands inside it. If you point at a tool withthe mouse, you will notice that its name

appears next to it, and what it is used for appears in the left bottom corner of the document.

To show the features of GSP, a series of examples follow. Each example will introduce a

different feature of the software in the context of constructing a geometric figure.

Toolbox

Selection Arrow Tool: Enables the selection of objects on screen.

Point Tool:Enables points to be added to screen.

Compass Tool:Enables circles to be added to screen.

Straight Edge Tool:Enables lines and rays to be added to screen.

Text Tool:Enables text to be added to screen.

CustomTool:Enables one to save diagrams as tool.

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GEOMETRY

Example 1. The three angles of a triangle sum to 180o.

1. Click on the Straight Edge tool [ ] and form the first line segment of the triangle.

2. Clickon one of the end points of the line segment to draw the next line segment.

3. Repeat step 2 to complete the triangle.

4. Select the sides of the triangle and go to the Display Drop Down Menu and select Line

Width and then select Thick. The triangle will now have thick lines. You can also

changethe colour of the lines by the same selections

5. You can label the vertices of thetriangle by selecting the point at each vertex, right

clicking and selecting Label Point. When you do this the following dialogue box

appears.

6. You can choose any label for the point and you can choose any font size using this

dialogue box. Click OK when finished.

7. To measure the angles in the triangle select the points in order, selecting the point at

which the angle is at as the middle click. Go to the Measure Drop Down Menu and

select Angle. The measure of that angle will appear on the screen as,

for example mABC = 43o

8. Measure the other two angles in the same way.

9. Go to the Measure Drop Down Menu and select Calculate. The calculator will appear.

10. Click on for example mABC = 43othis will appear on the calculator screen. Click +

on the calculator and then mACB, then + and mBAC. Then click OK on the

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calculator. The sum of the three angles will sum to 1800.

11. Select one vertex of the triangle and by holding down the Left Mouse key you can

change the orientation of the triangle. This will show that regardless of the orientation

of the triangle the sum of the angles will be 1800. See Diagram of below.

NOTE: Sometimes the angles do not add to 1800when you look at the individual angles

in the triangle. This is due to the decimal place inaccuracy. This can be change by going

to Preferences in the Edit Drop Down Menu. You can set the decimal place values to get

greater accuracy.

Example 2. Tangent to a circle and animation of the tangent.

1. Click on the Compasstool [ ] and form a circle on the screen by holding down the

Left Mouse Key and dragging outward from the point.

2. Click on Point tool and form a point on the circle.

3. Go back up the Selection Arrow tool [ ] and select the point you have formed on

the circle and the centre point of the circle.

mABC+mACB+mCAB= 180

mCAB= 55

mACB= 64

mABC= 61

B

A

C

Diagram for Example 1

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3. Repeat the last procedure taking B as the starting point and A as the ending point.

Another circle of the same radius will be drawn.

4. Using the Selection Arrow tool [ ] select the two circles (make sure nothing else is

selected). Go to the Construct Drop Down menu and select Intersection. The points of

intersection of the circles will be shown. Label the one above [AB] as C.

5. Select in turn, the two circles and the point below [AB], right click each time and select

Hide Circle and Hide Point.

6. You should be now left with three points A, B and C.

7. Click on the Straight Edge tool [ ] and draw a line segments between these points.

8. The triangle formed should be an equilateral triangle, which you can check in two ways.

Method 1. Go to the Measure Drop Down menu and measure the angles.

Are they all 60o?

Method 1. Go to the Measure Drop Down menu and measure the lengths of the sides.

Are they all equal?

m AB= 7.57cm

m CB= 7.57cm

m CA= 7.57cm

mCBA= 60

mACB= 60

mCAB= 60

C

A B

Diagram for Example 3

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Example 4. Constructing a right triangle.

1. In order to draw an accurate diagram it is best to set the length measurements correct to

two decimal places. To do this, go to the Edit Drop Down menu and select preferences.

The following dialogue box will appear.Click OK when finished.

2. Click on the Straight Edge tool [ ] and draw a line segment. Label the endpoints, say A

and B.

3. Again using the Selection Arrow tool [ ] select the line segment and one of the end

Points ( say B).Go to the Construct Drop Down menu and select Perpendicular Line.

4. Using the Point tool [ ] place a point on this perpendicular line. Label as C.

5. Using the Selection Arrow tool [ ] select theperpendicularline, right click and Hide

this perpendicular line.

6. Using the Construct Dropdown menu construct a line segment using the points B and C.

7. Join C to A to complete the right angled triangle.

This right angled can be saved in the Tool Box as follows and then you can use it again when

you want to draw a right angled triangle without going through the whole process again.

1. Go to the Edit Drop Down menu and the use Select All. The right angled triangle will be

selected.

2. Click on and hold down the small arrow of the Custom tool[ ] and select Create New Tool.

The following dialogue box will appear.

You can name this tool by typing into the box

And then clicking OK.

Set this to hundredths

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3. To reuse this tool (right angled triangle) Click on and hold down the small arrow of the

Custom tool select the tool and click and drag cursour across the screen and a right angled

triangle will be drawn.

Example 5. Constructing a circumcircle and shading in various parts of the diagram.

1. Construct a triangle as shown before.

2. Select one of the sides and going to the Construct Drop Down menu select mid-point.

3. Select this mid-point and its line and go to the Construct Drop Down menu select

Perpendicular Line.

4. Repeat steps 2 and 3 above for another side of the triangle.

5. Select the two perpendicular lines and go to the Construct Drop Down menu selectIntersection. This point is the circumcentre.

6. Select the two perpendicular lines and go to the Display Drop Down menu select

Hide Perpendicular lines.

7. Select the circumcircle and one of the vertices in that order. Go to Construct Drop Down

menu selectCircle by Centre + Point. The circumcircle will be drawn. This diagram is

shown below.

If you want to show shading in various parts of this diagram use the following method.

1. Select the three vertices of the triangle. Go to Construct Drop Down menu select

Triangle Interior. The triangle will now be shaded in. You can change the colour of the

shading by selecting the interior of the triangle, right clicking and selecting colour.

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2. If you want to shade in the circle. Select the circumference, go to Construct Drop Down

menu and select Circle Interior. You can change the colour of the shading again by right

clicking which gives you this option.

3. If you want to shade in one of the sectors a different colour you need to place points

close together on the circumference, select the points on the edge of the sector in correct

order and the go to Construct Drop Down menu to select Polygon Interior. See the series

of diagrams below.

4. Now go to Display Drop Down menu and Hide Points.

Points selected close together. New shaded area

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COORDINATE GEOMETRY:POINTS AND LINES

1. Open a new sketch. Go to Graph Drop Down menu, Grid Form and select Square Grid.The coordinate plane will appear on the sketch.

2. By holding down the Left Mouse key and dragging you can vary the scale.

Example 1. Adding Points to the Plane.

1.A point can be added to the plane in two ways. (a) by selecting the Point tool [ ] and

placing the point on the screen by clicking (b) by going to the Graph Drop Down menu

and selecting Plot Points. The coordinates of the points you want to add can be entered

in the dialogue box. NOTE; Using Method (a) the point can be dragged to any position

but using method (b) the point is fixed to the plane.

2. When a point is plotted, by selecting and right clicking you get a series of options as

shown in the diagram below.

NB. Select and Drag this point out to vary the scale.

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3. Plot two points on the plane using the Graph Drop Down menu. Select these points and go

to the Construct Drop Down menu you can select Line and this will draw the line going

through the two points. If you select this line and right click you are given the option offinding the Slope and Equation of this line. The only problem w ith this line is that it is static

so the slope and the equation are fixed. If we draw a line using a fixed point and a variable

point we can see relationships between various lines.

Example 2Showing slopes and equations of lines.

1. Plot two points on the plane one using the Graph Drop Down menu and the other using the

Point tool [ ]. Select these points and goto the Construct Drop Down menu you can

select Line and this will draw the line going through the two points. If you select this line

and right click you are given the option of finding the Slope and Equation of this line.

Make sure these measurement values are set to hundredths by using the Edit Dropdown

menu and Preferences. One of the points on this line ismovable so we can move the line to

various positions and see the equation and the slopechanging. This enables us to see the

relationship between the slopes of various lines and their equations clicking

Select this to show coordinates

Select this to show label

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Example 3. Point of intersection between two given lines.

1. Given the equations of two lines such asy = x + 4and 2x + 3y = 2. Plot each of these lines

using the Graph Drop Down menu and selecting New Function. The New Function dialogue

box will open as shown below.

.

2. Type in each function into the window. You must type each equation as a function so for

the first one type inx + 4 and for the second one type in (-2x+ 2)/3. Click OK after each

and the functions will be shown on the screen.

3. Select each function in turn and go to Graph Drop Down menu and select Plot NewFunction. The two lines will be drawn on the plane.

4. To find the point of intersection of thelines, select the Point tool [ ] and hover over the

point of intersection until the two lines are highlighted. Click in the intersection and the

point of intersection will be shown. You can label this point and find its coordinates by

selecting it andright clicking.

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Translations, Central Symmetry, Axial Symmetry and Enlargements.

Example 4. Translations

To show a translation let us take an example.

Find the coordinates of the image of the point (2,-1) under the translation (1,1) (5,2).

1. Open a new file.

2. Using the Graph Drop Down menuand selecting Plot Points plot the three points (2,-1),

(1,1) and (5,2) on the diagram.

3. Select the points (1,1) and (5,2) in that order go to the TransformDrop Down menu and

select Mark Vector. Select and join the points using the Construct menu.

4. Double click on the point you wish to translate, (2,-1), then select it and go to the Transform

Drop Down menu and select Translate the following dialogue box will appear.

5. Click on translate making sure that the Marked circle is ticked. The point will now be

translated. Join and label the points.

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Example 5. Symmetries

To show symmetries let us take an example.

Find the coordinates of the image of the point (3,4) under

(i) Axial symmetry in x axis Sx. (ii)Axial symmetry in y axis Sy

(iii) Central symmetry in the origin S(0,0). (iv) Central symmetry in the point (1,1) S(1,1).

1. Plot the point(3,4) as explained above.

(i) Double click on the x axis to mark as the axis of symmetry. Select the point (3,4) and go to

the Transform Drop Down menu and select Reflect. Right click to mark the coordinates of

the reflected point.

(ii) Double click on the y axis to mark as the axis of symmetry. Select the point (3,4) and go tothe Transform Drop Down menu and select Reflect. Right click to mark the coordinates of


(iii) Double click on the origin (0,0) to mark as the centre of symmetry. Select the point (3,4)

and go tothe Transform Drop Down menu and select Rotate. A dialogue box will open

and type in180oor if it shows radians. Click rotate and the point will berotated by 180o

which is the sameas reflecting in a point. Right click to mark the coordinates of the

reflected point.

(iv) Plot the point (1,1)and double click on itto mark as the centre of symmetry. Select the

point (3,4) and go to the Transform Drop Down menu and select Rotate. A dialogue box

will openand type in180oor if it shows radians. Click rotate and the point will be rotated

by 180owhich is the sameas reflecting in a point. Rightclick to mark the coordinates of


Example 6. Enlargements

To show an enlargement let us take an example.

Show the image of a triangle ABC under an enlargement of scale factor 2. The centre of the

enlargement being O.

1. Open a new file.

2. Using the Straightedge tool[ ] construct a triangle and label the vertices ABC.

3. Using the Pointtool [ ] plot a point to the left of the triangle and label as O.

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4. Double click on the point O to mark as centre. Select the triangle ABC and its vertices. Go

to the Transform Drop Down menu and select Dilate. The dialogue box below will appear.

Type in a Fixed Ratio of 2 : 1 and click on Dilate. The image triangle will be fromed.

5. To show the construction lines go to the Straightedge tool and hold down the small arrow

until you get the other choices [ ]. Click on the Ray [ and then click on the

point O, then A and release. Do the same for the point O and the points B and C.

Enlargement of ABC. Scale factor 2

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Example 1 Showthatthe circlesx2+ y2+ 2x + 2y7 = 0andx2+ y2- 2x + 4y -4 = 0have

two points of intersection. Hence find the length of the common chord.

1. The circlex2+ y2+ 2x + 2y7 = 0has centre (-1,1) and radius 3 and the circle

x2+ y2- 2x + 4y -4 = 0 has centre (1,-2) and radius 3 and the circle.

2. Draw the two circlesby putting in the centres and two a point on each circle and using

the Construct Drop Down menu and Circle by Centre + Point.

3. To show that the two circles cut in two places select both circles and using the

Construct Drop Down menu and Intersection the two pints are shown.

4. Select both points and using the Construct Drop Down menu and Segment draw the

line segment between the two points.

5. Select the Measure Drop Down menu to measure the length of the line segment.

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Example 2 Find the equation of the image of the circlex2+ y2+ 4x6y 12 = 0under the

translation (5,2)(10,1)

1. The circlex

2

+ y

2

+ 4x

6y

12= 0has centre (-2,3) and radius 5.2. Draw the circle by putting in the centre and a point(3,3) which is on the circle, then

using the Construct Drop Down menu and Circle by Centre + Point.

3. Usingthe Graph Drop Down menu and selecting Plot Points plot the points (5,2) and

(10,1).

4. Select the points (5,2) and (10,1) in that order go to the Transform Drop Down menu

and select Mark Vector. Select and join the points using the Construct menu.

5. Select the circle and go to the TransformDrop Down menu and select Translate and

the following dialogue box will appear.

6. Click on Translate making sure that the Marked circle is ticked. The image circle will

now be appear.

.

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