Geogebra #6
Name_________________________________________
Turn off your grid and axes for this activity. Label each point as directed (you can right-click on any point and RENAME it).
Save all 4 constructions by going to SAVE AS… -> COMPUTER-> PUBLIC-> CLASS FOLDER-> KSCOTT-> your class period.
1) Construct any ABC using tool 5A (you do not have to create the points first, just click on 3 random locations, then back to point A)
BEFORE YOU BEGIN THE INVESTIGATION, measure each angle of the triangle using tool 8A
2) Using tool 4A, construct a line that is perpendicular to
intersection of the perpendicular line and
AC
AC
as point D.
through point B. (select point B, then
AC ).
Using tool 2D, label the
BD is called the ALTITUDE of AC
Define an ALTITUDE of a triangle:____________________________________________________________________________________
2a) Use tool 1A to drag point B so that all angles are acute. Is
2b) Drag B until
A
is a right angle. Is
2c) Now drag point B until
A
BD
BD
located inside, outside or on one side of ABC ?________________
located inside, outside or on one side of ABC ?________________
is an obtuse angle. Is
BD
located inside, outside or ON
ABC ?________________
ABC have?____ Construct all other altitudes, then use tool 2D to label the altitudes’ intersection as E.
Point E is the orthocenter of ABC . Drag the vertices to create each type of triangle below. Determine if the orthocenter is inside,
outside, or ON ABC :
3a) When ABC is an acute triangle, the orthocenter is located ______________the triangle.
3b) When ABC is a right triangle, the orthocenter is located on ______________of the right angle
3c) When ABC is an obtuse triangle, the orthocenter is located ______________the triangle.
3) How many altitudes does
**Save this sketch as your name followed by the title ORTHOCENTER. SAVE it in our class folder
For Constructions 2-4, open a new file and redraw
4) Using Tool 2E, find the midpoint of
AC
ABC
(or “undo” until you arrive back to the original triangle with each angle measured)
and draw a segment (tool 3B) from B to midpoint D.
BD is the MEDIAN of AC .
Define a MEDIAN of a triangle:_______________________________________________________________________________________
4a) Use tool 1A to drag point B until all angles are acute. Is
4b) Drag the point B until
4c) Now drag point B until
A
A
5) How many medians does
is a right angle. Is
BD
is an obtuse angle. Is
BD
located inside, outside or on a side of ABC ?____________________
located inside, outside or on a side of ABC ?____________________
BD
located inside, outside or on a side of
ABC ?_________________
ABC have?_____Construct all other medians, then use tool 2D to label their intersection point G.
Point G is
the centroid of ABC . Drag the vertices to create each type of triangle below. Determine if the centroid is inside, outside, or ON ABC :
5a) When
5b) When
5c) When
ABC is an acute triangle, the centroid, G, is located ________________the triangle.
ABC is a right triangle, the centroid, G, is located_________________the triangle.
ABC is an obtuse triangle, the centroid, G, is located ________________the triangle.
**Save this sketch as your name followed by the title CENTROID. SAVE it in our class folder
6) Use the corner of a notecard as a “right angle tool”
Construct the orthocenter of
ABC
7) Measure the length of each side of DEF to the nearest .1 cm.
Label the midpoint of each side using your ruler.
Construct the centroid of DEF .
8) Using Tool 4D, construct the angle bisector of
(Select the tool, then click on points A, then B, then C in counterclockwise order)
With tool 2D, find the intersection of the angle bisector and
AC
and label that point D.
BD
is the ANGLE BISECTOR of
B .
Define an ANGLE BISECTOR of a triangle:_____________________________________________________________________________
8a) Use tool 1A to drag point B until all angles are acute. Is
8b) Drag the point B until
8c) Now drag point B until
A
A
or
or
C is a right angle.
Is
BD
BD
located inside, outside or on a side of ABC ?____________________
located inside, outside or on a side of ABC ?____________________
C is an obtuse angle. Is BD located inside, outside or on a side of ABC ?__________________
ABC have?____ Construct all other angle bisectors then use tool 2D find their intersection. Rename
9) How many angle bisectors does
the intersection point H. Point H is called the incenter of this triangle. Drag the vertices to create each type of triangle below. Determine if
the incenter is inside, outside, or ON ABC :
9a) When ABC is an acute triangle, the incenter, H, is located ______________the triangle.
9b) When ABC is a right triangle, the incenter, H, is located ________________the triangle
9c) When ABC is an obtuse triangle, the incenter, H, is located _______________the triangle
The term incenter is given to this point because it is the center of the circle that is contained inside this triangle. To find radius of this circle,
we must draw a perpendicular line from point H to one side of the triangle. This segment represents the DISTANCE from H to each side.
10a)
10b)
10c)
10d)
Select tool 4A and construct a perpendicular line from point H to AC . (click H, then select AC )
Use tool 2D to find the intersection of this perpendicular line and the side of the triangle. Label it point E.
Use tool 6A to draw a circle whose center is H that goes through point E. The circle should fit “snuggly” inside the triangle.
Use your arrow tool (1A) to move points A, B, and C. The circle should remain contained inside your triangle.
11) Use the intersection tool (2D) to find 2 other points where the circle intersects ABC . These intersection points will be labeled F and G.
Use tool 8A to record these distances: HE =____, HF = ____, HG = ____
12) Open a text box and record this statement, filled in with the correct TERM:
The distance from the incenter to each side of the triangle is always______________.
**Save this sketch as your name followed by the title INCENTER. SAVE it in our class folder
AC (You must select the tool first, then click on side AC )
With tool 2D, find the intersection of the perpendicular bisector and AC and label that point D. Does this line go through point
13) Using Tool 4C, construct the perpendicular bisector of
B?_________ Use tool 2A to identify on other point on this line. Label the point E. ED is the perpendicular bisector of
B .
Define: PERPENDICULAR BISECTOR of a triangle:___________________________________________________________________
11b) Use tool 8A to find the length of each side of your triangle. Use your arrow tool to move the vertex points to create a scalene,
isosceles, and equilateral triangle. For what type(s) of triangle does the perpendicular bisector go through a vertex point?____________
12) How many perpendicular bisectors does ABC have?____ Construct all other perpendicular bisectors then use tool 2D to find their
intersection. Rename this intersection point J. J is called the circumcenter of this triangle. Drag the vertices to create each type of triangle
below. Determine if the circumcenter is inside, outside, or ON ABC :
ABC is an acute triangle, the circumcenter, J, is located ______________the triangle.
When ABC is a right triangle, the circumcenter, J, is located ________________the triangle
When ABC is an obtuse triangle, the circumcenter, J, is located _______________the triangle
12a) When
12b)
12c)
The term circumcenter is given to this point because it is the center of the circle that “surrounds” the triangle. This circle contains all the
vertices of ABC . The radius of this special circle will be the circumcenter point, J, and one vertex of the triangle.
13a) Use tool 6A to draw a circle whose center is J that goes through one vertex point (A). All 3 vertices of the triangle should now be
located on the circle.
13b) Use your arrow tool (1A) to move points A, B, and C. The triangle vertices should always remain on your circle.
14) Use tool 8A to measure the distance from the circumcenter, J, to each vertex of the triangle. JA = _____, JB = _____, JC = _____
Open a text box and fill in the following, filled in with the correct TERM:
The distance from the circumcenter to each vertex of the triangle is always______________.
**Save this sketch as your name followed by the title CIRCUMCENTER. SAVE it in our class folder