NAME______________________
DATE_________
SKETCHPAD ACTIVITY: POINTS OF CONCURRENCY
Construct a large triangle that will be used for each of the following parts.
- Use the segment tool on the left toolbar.
Find the measure of each of the angles of the triangle.
Part I: Angle Bisectors.
Define Angle Bisector:
Change the color for the next items by going under the Display Toolbar.
Construct the three angle bisectors of the triangle.
- Using the Pointer Tool, highlight the three vertices of one of the angles,
then under the Construct Toolbar, click on Angle Bisector.
- Repeat the above step for the other two angles of the triangle.
Complete the following sentence:
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The three angle bisectors of a triangle intersect at __________ point.
Note: this point is called the INCENTER.
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Plot a point at the intersection of the three angle bisectors by highlighting two
of the angle bisectors, then under the Construct Toolbar select Point at
Intersection. Label it I for “Incenter”.
-
Measure the distances from the incenter to the sides of the triangle.
- Highlight the incenter and a side, then go to the Measure menu and select
distance
- Repeat for the other sides
Manipulate the triangle by dragging any of the sides or vertices. What happens
to the incenter? To the distances? How is this useful?
-
**Print your angle bisectors and incenter.
-
Highlight the three angle bisectors and hide them by going under Display and
selecting Hide Rays. Then highlight the incenter and under Display select Hide
Point.
Part II: Perpendicular Bisectors
Define Perpendicular Bisector:
-
Under Display, change the color to something different than Part I.
Construct the three perpendicular bisectors of the triangle.
- Highlight one of the sides of the triangle, then under the Construct Toolbar,
select Point at Midpoint.
- Highlight both the midpoint and the side it is on, then under the Construct
Toolbar select Perpendicular Line.
- Repeat the above steps for the other two sides of the triangle.
-
The three perpendicular bisectors of a triangle intersect at ___________
point.
Note this point is called the CIRCUMCENTER
-
Plot a point at the intersection of the three perpendicular bisectors by
highlighting two of the perpendicular lines and selecting Point at Intersection
under the Construct Toolbar. Label it C for “Circumcenter”.
-
Measure the distances from the Circumcenter to the Vertices of the triangle.
- Highlight the circumcenter and a vertex, then go to the Measure menu and
select distance
- Repeat for the other sides
-
Manipulate the triangle by dragging any of the sides or vertices. What happens
to the circumcenter? ...to the distances? How is this useful?
**Print your construction.
- Hide the three perpendicular bisectors and the circumcenter as you did in Part I.
Part III: Altitudes
Note: The Altitude of a triangle is a line that is
perpendicular to a side of the triangle
and goes thru the vertex opposite the side.
-
Under Display, change the color to something different than the first two
parts.
Construct the three altitudes of the triangle.
- Highlight one of the vertices and the side opposite to that vertex, then
under Construct select Perpendicular Line.
- Repeat the step for the other two vertices.
The altitudes of a triangle meet at _______ point.
Note: We call that point the ORTHOCENTER.
-
Plot a point at the intersection of the three altitudes. Label it O for
“Orthocenter”.
-
Manipulate the triangle by dragging any of the sides or vertices. What happens
to the orthocenter?
**Print your Construction.
-
Hide the altitudes and the orthocenter.
Part IV: Medians
Note: The median of a triangle is a segment
that connects a vertex of the triangle
to the midpoint of its opposite side.
Select a new color under the Display Toolbar.
Construct the three medians of the triangle.
- Highlight a side and under Construct select Point at Midpoint.
- Using the Segment Tool, construct a segment from the midpoint to the
opposite vertex of the triangle.
- Repeat the above steps for the other two sides.
Holy Cow!! The medians intersect at _______ point.
-
Note: We call this point the CENTROID.
-
-
Plot a point at the intersection of the medians. Label it M even though it is the
“Centroid”.
Measure the length of the median.
- Highlight the Median go to the Measure menu and choose Length.
Now measure the two parts of the median.
- Highlight the centroid and the vertex go to the Measure menu and select
Distance
- Repeat for the distance between the Midpoint and the centroid.
Repeat this process for the other two medians.
Manipulate the triangle by dragging any of the sides or vertices. What happens
to the centroid? What do you notice about the length of the meadian and its
parts?
** Print your construction.
Part V: All Four Points of Concurrency
- Under Display select Show All Hidden. You now should have all four “points of
concurrency” showing on the same triangle.
- Manipulate the triangle by dragging any of the sides or vertices. Do the points
of concurrency ever intersect or coincide? When? For what type of triangles?
Where?