Welcome to the Spoken tutorial on Properties of Circles in GeoGebra.
In this tutorial, we will learn about the properties of, Chords
Arcs and sectors and Tangents
To record this tutorial, I am using;
Ubuntu Linux OS version 18.04
GeoGebra version 5.0.660.0-d
The steps demonstrated in this tutorial will work exactly the same in lower versions of GeoGebra.
To follow this tutorial, learner should be familiar with GeoGebra interface.
For the prerequisite GeoGebra tutorials please visit this website.
I have opened a new GeoGebra window.
Let us uncheck the Axes.
Right-click in the Graphics view.
In the Graphics menu, uncheck the Axes check box.
In theAlgebra view click on the Toggle Style Bar arrow.
In the Sort by drop-down, select Object Type check box, if not already selected.
Let us now learn about the property of a chord.
It states that - Perpendicular from the centre of a circle to a chord bisects the chord.
Let us draw a circle. Select the Circle: Center & Radius tool from the tool bar.
Click in the Graphics view to mark a point A.
Circle: Center & Radius text box opens.
In the Radius field let us type 3 and click the OK button.
A circle c with centre A and radius 3 centimetres is drawn in the Graphics view.
Select the Segment tool.
Click to mark two points B and C on the circumference as shown.
Chord BC, named as f is drawn on the circle c.
Let’s drop a perpendicular line to chord BC passing through A.
Click on the Perpendicular Line tool.
Click on chord BC, and then on point A.
Let us move point B.
Observe that the perpendicular line moves along with point B.
The perpendicular line and chord BC intersect at a point.
Using the Intersect tool let’s mark the intersection point as D.
Let’s measure the lengths BD and DC.
Click on the Distance or Length tool.
Click on the points, B and D and then D and C.
Notice that distances BD and DC are equal.
It implies that D is midpoint of chord BC.
Note that the perpendicular from the centre A to chord BC bisects it.
Let us move all the labels using the Move tool to see them clearly.
Now let’s measure the angle CDA.
Click on Angle tool and click the points C, D and A.
Angle CDA is 90 degrees.
A line drawn from the centre to the midpoint of the chord is perpendicular to it.
Let us move point C and see how the distances change accordingly.
Pause the tutorial and do this assignment.
Open a new GeoGebra window.
Draw a circle.
Draw two chords of equal size to the circle.
Draw perpendicular lines from the centre to the chords.
Mark points of intersection.
Measure the perpendicular distances.
What do you observe?
The completed assignment should look like this.
Observe that, equal chords of a circle are equidistant from centre.
Now let us go back to the circle.
Let us retain circle c and points A, B and C.
Delete the rest of the objects.
Go to the Algebra view.
Press the Ctrl key and select the objects for deletion.
Then press Delete key on the keyboard.
Next let us prove a property with respect to an arc.
Inscribed angles BDC and BEC subtended by the same arc BC are equal.
Let us next draw an arc.
Click on the Circular Arc tool.
Click on point A.
Then click on points B and C on the circumference.
An arc d is drawn.
Let us change properties of arc d.
In the Algebra View, right-click on object d.
Select Object Properties from the context menu.
Properties window opens next to Graphics view.
Click on the Color tab and select green colour.
Let us change the style of filling of the arc d.
Select the Style tab and change the Filling to Hatching.
Close the Properties window.
Let us mark two points on the circumference of the circle.
Click on Point tool.
Mark point D above point B and point E above point C.
Let us subtend two angles from arc BC to points D and E.
Select the Segment tool and join the following points.
B,E E,C B,D and D,C.
Let’s measure the angles BDC and BEC.
Click on the Angle tool,
Click the segments that form the angle.
BD and DC and then click BE and EC.
Observe that the angles BDC and BEC are equal.
This proves the property that angles formed using the same arc are equal.
Let’s draw a sector ABC.
Click on Circular Sectortool.
Now click the points A, B, and C.
Sector ABC is drawn.
Let’s measure the angle BAC using the Angle tool.
Observe that angle BAC is twice the angles BDC and BEC.
Using the Move tool let’s move point C to change the angles.
Notice the angles BEC and BDC subtended by the arc d.
Angle BAC is always twice the angles subtended by the arc d.
Here angle at the centre is twice any inscribed angle subtended by the same arc.
Next let us construct a pair of tangents to a circle.
Let us open a new GeoGebra window.
Let us uncheck the Axes.
Let's draw a circle using Circle: Center & Radius tool.
Click in the Graphics view to mark point A.
Type 3 for radius in the text box that opens.
Then click OK button.
A circle c with centre A and radius 3 centimetres is drawn.
Now click on the Point tool.
Click to mark a point B outside the circle.
Using the Segment tool join points A and B to draw segment f.
Let us draw a perpendicular bisector to segment f.
Select the Perpendicular Bisector tool, click on points A and B.
Segment f and perpendicular bisector intersect at a point.
Click on Intersect tool to mark the point of intersection as C.
Let's move point B.
Observe that perpendicular bisector and point C move along with point B.
This is because these objects are dependent on point B.
Pause the tutorial and do this assignment.
Verify if point C is the midpoint of segment f.
Now let us draw another circle.
Select the Compass tool.
Click on the points C, B and C again to complete the figure.
Two circles intersect at two points.
Using the Intersect tool, mark the points of intersection as D and E.
Select the Segment tool.
Join the points B, D and B, E.
Segments h and i are the tangents to circle c.
Let's explore some more properties of the tangents to the circle.
Using the Segment tool and join the points A, D and A, E.
Let us show that triangles ABD and ABE are congruent.
Segment j is equal to segment k, as they are radii of circle c.
In the Algebra view observe that segment j is equal to segment k.
Angle ABD is equal to angle BEA (∠ADB = ∠BEA).
As they are angles on the semicircles of the circle d. Let’s measure the angles.
Select the Angle tool.
Click the segments j, h and i, k to measure the angles.
Notice that are equal and 90 degrees.
Segment f is the common side for both the triangles.
Therefore triangle ABD is congruent to triangle ABE by SAS rule of Congruence.
It implies that tangents BD and BE are equal.
From the Algebra view, observe that segments h and i are equal.
Tangents are perpendicular to the radius of the circle at the point of contact.
Let's move point B and see how the tangents move along with point B.
Tangents are drawn from point B', so they are dependent on it.
Let’s now delete point B.
Right-click on point B, from the context menu select Delete.
All the objects dependent on point B are deleted along with it.
We now have a circle c with centre A on the Graphics view.
Select the Point tool.
Mark points B and C on the circumference and D outside the circle.
Select the Tangents tool.
Click on point D and then on the circumference.
Two Tangents are drawn to the circle c.
Tangents meet at two points on the circle.
Click on the Intersect tool and mark points of contact as E and F.
Let us draw a triangle.
Click on the Polygon tool.
Click on the points B, C, F and B again to complete the figure.
In the figure segment b is the chord to the circle c.
Angle FBC is the inscribed angle by the chord CF to the circle c.
Angle DFC is the angle between tangent and chord to circle c.
Let’s measure the angles.
Click on the Angle tool.
Click on the points F, B, C and D, F, C.
Notice that angle DFC is equal to angle FBC .
Angle DFC is the angle between tangent and chord CF.
This angle is equal to inscribed angle FBC of the chord CF.
Let's move point D. Observe that tangents and chord CF move along with point D.
Here all the objects are dependent on point D as the tangents are drawn from it.
Let us save this file now
Click on File then Save.
I will save the file on the Desktop.
In the Save dialog box type the file name as Tangents. Click on Save button.
With this, we come to the end of the tutorial. Let us summarise.
In this tutorial, we have learnt about the properties of, Chords, Arcs and sectors and Tangents
As an assignment. Open a new GeoGebra window.
Draw a circle.
Draw tangents from an external point.
Mark points of intersection of the tangents.
Join the centre of the circle to intersection points
Measure angle at the centre and measure angle between the tangents.
What is the sum of the two angles?
Join the centre and the external point.
Does the line segment bisect the angle at the centre?
The output of the assignment should look like this.
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This is Madhuri Ganapathi from, IIT Bombay signing off. Thank you for watching.