Analytic Geometry on GeoGebra:  Centres of a Triangle

 

Instructions

 

Click on the link for the Analytic Geometry on GeoGebra Applet.  A new window will open with a GeoGebra applet which you will use.  You need to have the Java plug-in installed on your machine to run this applet.  If you don’t, go to www.java.com and download it.  It is free and will not harm your computer.

 

When you see the window, notice that there are several different tool categories at the top of the page.  Several have little arrows on the bottom right of the icon.  If you click the arrow on each, you are shown other related tools which you will be using to complete the exercises.  When you mouse on a tool, and wait a few seconds, you will see the name of the tool.

 

On the top right of the applet is a refresh button which will re-set the applet to its original state.

 

Make sure you learn the concepts you demonstrate.  Make notes on the definitions and summaries I provide.  Most of all have fun!

 

Part 1:  The Basics

§         Click on the tool “Segment between two Points.” 

§         Make a line segment anywhere on your grid by clicking for the beginning point of your line segment, and clicking for the end point of your line segment.  Notice that on the left panel, an Algebra Window will keep track of your points, and later your calculations.

§         Select the tool, Distance or Length, then click on your line segment, and it will measure your line in grid units.  It should appear on your graph and in the Algebra window on the left.

§         Select the arrow on the bottom right of the Distance tool, and more tools will appear.  Select the Slope tool, then click on your line, and it will tell you the slope of your line segment.

§         Select the tool, Midpoint or Centre, and then your line and it will put a point at the centre of your line segment.  The coordinates of your midpoint will be stated in the Algebra window on the left.

 

Part 2:  The Basics, Part 2

§         Refresh the applet with the refresh button on the top right of the applet.

§         Make a new line across most of the grid.

§         Select the New Point tool and make a new point on the grid away from your line segment.

§         Select the Perpendicular line tool.  Click on the point, and then on the line segment.  A line perpendicular to your line segment and through your point will appear.

§         Find the Parallel Line Tool (under the perpendicular tool).  Click the point, and then a line, parallel to your first line and through your point will appear.

§         Find the Perpendicular Bisector tool.  Click on any line.  A line which is perpendicular to that line and through its midpoint will appear. 

ü      Summary:  Perpendicular = 90o, and bisector = midpoint.

 

Part 3:  Triangle Medians and Triangle Centroids

§         Refresh the applet by using the reset button.

§         We are going to make a large triangle.  Select the Polygon Tool.  Click on the grid three times, once for each vertex (corner) of your triangle, then click a fourth time on the very first vertex you made.  We are now going to construct the three MEDIAN lines of this triangle.  Median = midpoint to opposite vertex.

§         Select the Midpoint tool, and click on each side of the triangle.  The midpoint of each side will appear.

§         Select the Line Segment tool and join each midpoint to its opposite vertex (do not join the midpoints to each other).

§         The three median lines will intersect at one point.

§         Select the Intersect Two Objects tool.  Click on one median line and then a second median line.  A new point will appear at the intersection of the median lines.  This point is called the CENTROID.

§         Now, select the Arrow tool, and click on one of your vertices.  Drag the vertex around and notice that no matter what shape the triangle takes, the property of medians intersecting at the CENTROID always holds true.  Also, the centroid is always either inside the triangle, or on one of its sides.

ü      Summary:  The intersection of the Medians of a triangle is called the CENTROID.  The centroid is the centre of the triangle’s mass.

 

Part 4:  Triangle Altitudes and Orthocentres

§         Refresh the applet, and make a new triangle using the Polygon tool.

§         An altitude is a line from a vertex to its opposite side which is perpendicular to that opposite side. 

§         Find and select the Perpendicular line tool.  Click on a vertex, and then its opposite side.  The altitude will appear.  Do this for the other sides, and notice again that the three altitudes all go through the same point.  This point is called the ORTHOCENTRE. 

§         Select the Arrow tool, and click and drag on any vertex and notice that no matter where you move the vertex, the altitudes will all intersect at the same point.  Also notice that it is possible for altitude lines and the ORTHOCENTRE to be outside the triangle.  Drag a vertex to make an obtuse angled triangle, and you will see it for yourself.

ü      Summary:  The intersection of the Altitudes of a triangle is called the ORTHOCENTRE. 

 

 

Part 5:  Perpendicular Bisectors and Circumcentres

§         Refresh the applet, and make a new triangle.

§         We are now going to make Perpendicular Bisectors using the perpendicular bisector tool.  Find and select this tool, and then click on each side of the triangle.  Remember, Perpendicular = 90o, and bisector = midpoint.

§         Once again, the perpendicular bisectors all intersect at a single point.  This point is called the CIRCUMCENTRE.

§         Find the Intersect Two Object tool, and select it.  Then click on two perpendicular bisectors and a point should appear at the circumcentre.

§         This part is new:  Find the Circle with a Centre through Point tool.  Click on the triangle’s circumcentre and drag out towards a vertex.  You should see a circle being formed.  Drag this circle to a vertex.  You should notice that all three vertices are on the circle. 

§         Select the Arrow tool, and click and drag a vertex and you will see that the property holds no matter what shape the triangle takes.  As well, the circumcentre as well as the perpendicular bisectors can leave the middle of the triangle

ü      Summary:  The intersection of the perpendicular bisectors of a triangle is called the CIRCUMCENTRE.  A circle can be made with the circumcentre as its centre, which intersects with all three vertices.  This means that the circumcentre is the centre of the vertices of the triangle, and that the distances from each vertex to the circumcentre are all equal.

 

 

Part 6:  Angle Bisectors and the Incentre

§         Refresh your applet, and make a new triangle.

§         Find the Angle Bisector tool, and one at a time select each angle in your triangle to bisect.  Remember, it takes three points to define an angle so you need three clicks to identify and bisect one angle.

§         The angle bisectors all intersect at a point.  Select the Intersect Two Objects tool, and make a make a point at their intersection.  Then select the Circle tool and make a circle centred at the incentre.  Notice that this circle will touch all three sides at the exact same time.  Select the arrow tool, and move the vertices to show that this property holds true for all triangles.  This circle is called an INCIRCLE because it is inscribed perfectly in the triangle and intersects each of the triangle’s sides once (tangent to each side).

ü      Summary:  The intersection of the angle bisectors of a triangle is called the INCENTRE.  A circle can be made with the incentre as its centre, which is simultaneously tangent (just touches once) to all three of the triangle’s sides.  This means that the incentre is the centre of an inscribed circle of the triangle, called the INCIRCLE.

 

You are almost done!

 

Part 7:  Euler’s Line

§         Refresh your applet and make a new triangle.

§         Use what you have learned to find the Orthocentre, Centroid and Circumcentre of your triangle.

§         You should notice that each of these three points are collinear, meaning that they are on the same line.  Use the line tool to make the line through these three points.  Now select the Arrow tool, and drag the vertex of the triangle and you will see that these three centres are always in a line no matter what triangle you have.  A mathematician named Leonhard Euler (pronounced Oiler) found this line in the 18th century.

 

 

All of these triangle properties have their own application to design and engineering.  You will be doing a project in a few weeks which will make use of this information which simulates a real-live engineering task.

 

 

Wasn’t that fun?  Now that is Geometry in action!