Geogebra/C3/Tangents-to-a-circle/English
Title of script: Tangents to a circle in Geogebra.
Author: Neeta Sawant
Keywords: Tangents, point of tangency', 'Perpendicular Bisector', 'Intersect two Objects', 'Compass', 'Angle', 'Polygon', 'Circle with Center and Radius', inscribed angle, Chord Spoken tutorial, video tutorial.
Note to Translators - Translators do no translate theorems , please refer to standard mathematics text books of classes IX and X of your language for the original version of the theorems.
| Visual Cue | Narration |
|---|---|
| Slide Number 1 | Hello everybody.
Welcome to this tutorial on "Tangents to a circle in Geogebra". |
|
Slide Number 2 Learning Objectives |
At the end of this tutorial you will be able to
|
| Slide Number 3
Pre-requisites |
We assume that you have the basic working knowledge of Geogebra.
If not, For relevant tutorials on Geogebra, Please visit our website |
| Slide Number 4
System Requirement |
To record this tutorial I am using
Ubuntu Linux OS Version 11.10 Geogebra Version 3.2.47.0 |
| Slide Number 5
GeoGebra Tools used |
We will use the following Geogebra tools
|
| Switch to GeoGebra window
Dash home >>Media Apps>>Under Type >>Education>>Geogebra |
Let's open a new GeoGebra window.
Click on Dash home Media Apps. Under Type Choose Education and GeoGebra. |
| Let's draw Tangents to a circle
let's define tangents to a circle | |
| Slide 6
Definition of a tangent Show the finished figure |
Tangent is a line that touches a circle at only one point
The point of contact is called "point of tangency" |
| Right Click on the drawing pad >>
Graphic view box opens>> un-check on Axes>>Select Grid |
For this tutorial I will use "Grid" instead of "Axes"
Right Click on the drawing pad In the "Graphic view" uncheck "Axes" Select "Grid" |
| Select “Circle with Center and Radius” tool>>Mark point 'A' | First let's draw a circle.
Select “Circle with Center and Radius” tool Mark a point 'A' on the drawing pad |
| A dialogue box opens | A dialogue box opens. |
| Type value '3' for radius
Click OK Visually show that you can move point 'A', but the circle remains of the same radius. |
Let's type value '3' for radius
Click OK A circle with centre 'A' and radius '3' cm is drawn. Let's 'Move' the point 'A' and see that circle has same radius |
| Click on "New point" tool >> Mark point 'B' | Click on "New point" tool
Mark a point 'B' outside the circle |
| "Select Segment between two points" tool>> join points 'A' and 'B' | "Select Segment between two points" tool.
Join points 'A' and 'B'. Segment AB is drawn |
| Select "Perpendicular Bisector" Tool >> Point A >> point B
Move the point B and show how the perpendicular bisector moves along with B. |
Select "Perpendicular Bisector" tool
Mark points 'A' and then 'B' A perpendicular bisector is drawn Let's Move the point 'B' and see how the 'perpendicular bisector' moves along with 'B'. |
| Click on "Intersect two objects" tool>>Point C
Again, move point 'B', and show that 'C' moves accordingly. |
Segment 'AB' and Perpendicular bisector intersect at a point
Click on "Intersect two objects" tool Mark point of intersection as 'C' Let's Move point 'B', and see how point 'C' moves along with 'B' |
| Click on "Distance" tool>>click on points 'A' and 'B'
|
How to verify whether 'C' is the midpoint of 'AB'?
Click on "Distance" tool. click on points 'A' , 'C'. and 'C' ,'B' Notice that 'AC' = 'CB' |
| Select "Compass" tool from tool bar>> Point 'C'>>Point 'B'>>Point 'C' | Select "Compass" tool from tool bar.
Click on points 'C', 'B'. and 'C' once again... to complete the figure. |
| Click on "Intersect two objects" tool>>Point 'D'>> Point 'E'
Point to the two points of intersection |
Two circles intersect at two points
Click on "Intersect two objects" tool Mark the points of intersection as 'D' and 'E' |
| Select "Segment between two points" tool>>Join 'B' and 'D' >>join B' and 'E' | Select "Segment between two points" tool
Join points 'B' and 'D'. 'B' and 'E' . |
| Point to the circle 'c' | Can you see that the Segements 'BD' and 'BE' are tangents to the circle 'c'? |
| Now, let's explore some of the properties of these Tangents to the circle | |
| Click "Segment between two points" tool>> join 'AD'>>join 'AE' | Select "Segment between two points" tool
join points 'A', 'D' and 'A', 'E' Segment 'AD'=Segment 'AE' (radii of circle 'c'). |
| outline the triangles 'ABD' and 'ABE'.
Point to Segments 'AD' and 'AE'
|
In triangles 'ABD' and 'ABE'
Segment 'AD'= segment 'AE' Let's check from Algebra view |
| Point to angles 'ADB' and 'BEA'
Outine the semicircle 'd'. |
'∠ADB'= '∠BEA' = '90°' (angle of the semicircle of circle 'd')
Lets check with "Angle" tool |
| Click "Angle" tool>>click on points >> ADB and BEA | Click on the "Angle" tool...
Click on the points 'A', 'D', 'B' and 'B', 'E', 'A' |
| Point to segment 'AB'
Outline the two triangles |
Segment 'AB' is common side for both the triangles
therefore '△ABD' '≅' '△ABE' by "SAS rule of conguency" |
| Point to tangents 'BD' and 'BE' | It implies that Tangents 'BD' and 'BE' are equal! |
| point to the Algebra view | From the Algebra view,
we can find that the tangents 'BD' and 'BE' are equal |
| Point to the Angle,Radius, Tangent | Please Notice
A tangent is always at right angles to the radius of the circle where it touches |
| Let us save this file now
Click on “File”>> "Save As" I will type the file name as "Tangent-circle" Click on "Save" |
Let us save this file now
Click on “File”>> "Save As" I will type the file name as "Tangent-circle" Click on "Save" |
| Let's state a theorem | |
| Slide 7
Theorem |
"Angle between tangent and chord at the point of tangency, is same as an inscribed angle subtended by the same chord". |
| Let's verify the theorem | |
| Click on “File” >> New | Let's open a new Geogebra window.
click on “File” >> "New" |
| Select "Circle with center through point" tool>>point 'A' >> point 'B' | Click on "Circle with center through point" tool.
Click on point 'A' as center, then on point 'B'. |
| Click on "New point" tool >> point 'C'>>Point 'D' | Click on "New point" tool.
Mark points 'C' on circumference of the circle and point 'D' outside the circle. |
| Click "Tangents" tool >>point D>> circumference | Click on "Tangents" tool.
click on point 'D'... and circumference. Two Tangents are drawn to the circle. |
| Point to 'E' and 'F'
Click on "Intersect two objects" tool>>Mark points of intersection |
The tangents intersect at two points on the circle.
Click on "Intersect two objects" tool Mark points of intersection as 'E' and 'F'. |
| Click on "Polygon" tool>>Click on point B>> point C>>point F>>point B again to complete the figure | Let's draw a triangle.
Click on the "Polygon" tool. Click on the points 'B' 'C' 'F' and 'B' once again to complete the figure |
| Point to the segment 'BF' | In the figure segment 'BF' is the chord to the circle 'c' |
| Point to '∠FCB' and the chord | '∠FCB' is the inscribed angle by the chord to the circle 'c' |
| Point '∠DFB' and tangent | '∠DFB' is angle subtended by the tangent.and chord to the circle |
| Click on "Angle" tool >> Point 'F' 'C' 'B' >> ' point D' 'F' 'B' | Lets Measure angles
Click on "Angle" tool click on the points 'F' 'C' 'B' and 'D' 'F' 'B' |
| Show the angles | Notice that '∠FCB' = '∠DFB' |
| Hence the theorem is verified | |
| Click on “File”>> "Save As"
I will type the file name as "Tangent-angle" Click on "Save" |
Let us save this file now
Click on “File”>> "Save As" I will type the file name as "Tangent-angle" Click on "Save" With this we come to the end of this tutorial. |
| Summary | Let's summarize
In this tutorial, we have learnt to verify that "Two tangents drawn from an external point are equal" "Angle between a tangent and radius of a circle is 90^0" "Angle between tangent and a chord is equal to inscribed angle subtended by the chord " |
| Assignment | As an assignment I would like you to verify:
"Angle between tangents drawn from an external point to a circle, is supplementary to the angle subtended by the line-segment joining the points of contact at the centre". To verify the theorem Draw a circle Draw tangents from an external point Mark points of intersection of the tangents Join center of circle to intersection points Measure angle at the center, and Measure angle between the tangents What is the sum of the two angles? Join center and external point Does the line-segment bisect angle at the center? Hint - Use Angle Bisector tool |
| Show the output of the Assignment | The output of the assignment should look like this
Sum of the angles =180^0. The line bisects the angle |
| Slide number 8
Acknowledgement |
Watch the video available at
http://spoken-tutorial.org/What is a Spoken Tutorial It summarises the Spoken Tutorial project If you do not have good bandwidth, you can download and watch it |
| Slide Nubmber 9 | The Spoken Tutorial Project Team :
Conducts workshops using spoken tutorials Gives certificates to those who pass an online test For more details, please write to contact@spoken-tutorial.org |
| Slide number 10 | Spoken Tutorial Project is a part of the Talk to a Teacher project
It is supported by the National Mission on Education through ICT, MHRD, Government of India More information on this Mission is available at ttp://spoken-tutorial.org/NMEICT-Intro ] Script – contributed by Neeta Sawant from SNDT Mumbai Naration- Madhuri Ganpathi from IIT Mumbai Thank you for joining |
