Geometry/Trig II
Geometer’s Sketchpad
I.
Introduction
After signing on to the computer, open programs and select
.
This will open Geometer’s Sketchpad. There should be a blank white screen with
commands on the top:
and a tool bar to the left of the screen.
We will be using both.
II.
Creating a triangle
1. Select the point tool from the tool bar.
2. In the white field click with the mouse to create a point.
3. Create two more spaced out in the field for a total of three points.
4. Select the arrow tool from the tool bar.
5. The last point you created should be highlighted. You can tell from the pink
highlight. Click on a second point to highlight another point.
6. From the top menu select “Construct” to get a drop-down menu.
Select the option “Segment” from the list. A segment should have
been drawn connecting the two points and will be highlighted.
7. Click into the white field to deselect the segment. Click on another
set of points and draw the segment and the same for the last two points.
8. You should now have a triangle drawn.
9. Once a triangle is drawn you can then change its look by clicking on a vertex and
while holding down the mouse move the vertex around the field. Give it a try.
III.
Labeling the triangle
1. Select the three vertices.
2. From the top menu select “Display” and from the drop-down menu
select “Show Labels.”
3. The three vertices should be labeled “A”,”B”, and “C” in the order that
you selected them in.
4. If not, double click onto a label and a menu should appear that you can modify
the labeling.
IV.
Copying diagrams to MS Word
1. To select objects to copy either select each one at a time or left click into the top
left corner, hold down the left click, and drag to the bottom right.
2. To select all objects you can also hold down the Ctrl+A keys.
3. To copy the selected objects go to the top menu, select “Edit” and from the drowdown list select “Copy.” Or hold down Ctrl+C.
4. Open a word document.
5. Once open go to the top menu, select “Edit” and from the drow-down list select
“Paste.” Or hold down Ctrl+V. This should paste the triangle to a new word
document.
6. Once a triangle is drawn you can then change its look by clicking on a vertex and
while holding down the mouse, move the vertex around the field. Give it a try.
7. Create and paste to a word document: 1) your original triangle; 2) an acute
triangle; and 3) an obtuse triangle.
8. Save the word document as “lastname_geo1” into your documents.
V.
Creating Medians
1. Select the segment AB .
2. In the top menu click on “Construct” and choose “Midpoint” from
the drop-down menu.
3. Show the label of the midpoint(D).
4. To draw the median, select the midpoint, D, and the opposite vertex, C.
Construct the segment.
5. Construct the other two medians by repeating steps 1-4. AC should have
midpoint E and BC should have midpoint F.
6. Copy your Triangle with the medians to you Word document.
7. Answer the following questions:
1-a. What do you notice about the three medians?
1-b. If you make different shaped triangles do the medians still intersect?
1-c. Does the intersection ever move out of the interior of the triangle?
VI.
Measuring the medians
1. Label the intersection of the medians, “G”
2. Select the vertex A and the intersection G.
3. From the top menu select “Measure” and “Distance” from the drop-down menu.
On the upper left part of the white field should be the box “AG = …”
4. Select the intersection G and the midpoint F and calculate the distance between
the points. Under “AG = …” should appear “GF = …”
5. From the top menu select “Measure” and “Calculate” from the drop-down
menu. A pop-up menu will appear. We are going to calculate the ratio of
the two parts of the median AF . Click on the “GF = …” box, then the
division sign, ÷, and finally click on the “AG = …” box. A new box should
have appeared.
1-d. What is the calculation of the ratio of the two parts of the median?
1-e. Move the vertices around. How does the ratio change?
1-f. The intersections of the medians is called the centroid. What do you think is
the relationship between the segment from the midpoint to the centroid and the
vertex and the centroid. Check your hypothesis using the other two sides. Copy
your work to your word document.
VII. Creating Altitudes
1. Create a blank sketchpad by selecting “File” from the top menu. Then choose
“New Sketch”
2. Create a new acute triangle and label the vertices.
3. Select the vertex A and the opposite side BC .
4. From the top menu select “Construct” and select “Perpendicular Line”
from the drop-down menu.
5. Repeat steps 3 and 4 for vertex C and side AB . Mark the intersection of
the two altitudes.
2-a. Does it appear that the third altitude will pass through the intersection of the
other two altitudes? Check your answer. Copy the triangle with the altitudes to
your word document. (This would be a good place to save if you haven’t in a
while)
2-b. The intersection of altitudes is called the orthocenter. Can you move the
vertices around so that the orthocenter is outside the interior of the triangle?
What kind of triangle is formed? Copy this triangle to your word document.
2-c. Can you move the vertices of the triangle around so that the orthocenter is at
one of the vertices? If so, what kind of triangle is this? Copy the new triangle to
your word document.
6. Select the three altitudes. From the top menu select “Display” and select “Hide
Perpendicular Lines” from the drop-down menu. The three altitudes should
disappear but the orthocenter should remain.
7. Construct segments from each of the vertices to the orthocenter. This should
create three triangles.
8. Choose one of the new smaller triangles. Construct the altitudes for the smaller
triangle.
2-d. What is the orthocenter for the smaller triangle? Copy the diagram to the
word document.
2-e. What is the relationship between the orthocenters of the three smaller
triangles and the original triangle?
VIII.
1.
2.
3.
Constructing an Angle Bisector
Create a new sketchpad.
Create a new triangle and label it.
An angle is defined by the points. Select ABC by selecting points ABC in that
order. From the top menu select “Construct” and “Angle Bisector” from the dropdown menu. A ray should be formed from B to AC .
4. Construct the angle bisector for BAC . Mark the intersection “D.”
3-a. Does it appear that the third angle bisector will pass through the intersection
of the other two altitudes? Check your answer. Copy the triangle with the
altitudes to your word document.
3-b. If you move the vertices or sides do the angle bisectors still intersect?
3-c The intersection of the angle bisectors is called the incenter. Can you move
the vertices around so that the incenter is outside the interior of the triangle?
5. Select the angle bisectors and hide them.
6. Select the incenter and the segment AC . Draw a perpendicular line through the
incenter to the segment AC . Mark the intersection of the perpendicular and the
segment, “E”. Hide the perpendicular line when done.
7. Repeat for the segments AB and BC marking intersections “F” and “G”. Hide the
perpendicular lines when done.
8. Create segments from the incenter to the 3 sides. Select all three and from the
top menu select “Measure” and “Length” from the drop-down menu. Three
measurements will appear on the top left.
3-d. What do you notice about the length of all three segments? Move the
vertices around. How are the lengths affected?
9. Since all the segments are equidistant from the incenter, we can draw a circle
inside the triangle. Select the incenter and the point on AC . From the top menu
select “Construct” and choose “Circle by Center+Point” from the drop-down
menu.
3-e. Does the circle ever move outside of the triangle? Move the triangle around
to see if you can get any part of the circle outside the triangle. This is called an
inscribed circle since it is drawn with in the triangle. Copy your triangle and
inscribed circle to your word document.
IX. Constructing Perpendicular Bisectors
1. Start with a new sketchpad.
2. Create and label a triangle.
3. Find the midpoints of each of the segments.
4. Select one of the midpoint and the segment it is on. Construct the perpendicular
line.
5. Select another midpoint and the segment it is on. Construct the perpendicular.
Mark the intersection of the two perpendicular bisec.
4-a. Does it appear that the third perpendicular bisector will pass through the
intersection of the other two perpendicular bisector? Check your answer. Copy
the triangle with the perpendicular bisector to your word document.
4-b. If you move the vertices or sides do the perpendicular bisector still intersect?
4-c The intersection of the angle bisectors is called the circumcenter. Can you
move the vertices around so that the circumcenter is outside the interior of the
triangle? If so copy it to your word document.
6. Hide the Construct segments from the circumcenter to each of the vertices. Find
the lengths.
4-d. What do notice about the lengths? Move triangle around and check the
distances.
7. Select the circumcenter and one of the vertices. Create a circle.
4-e. What do you notice about the circle and the triangle. Move the triangle
around. Does the triangle ever move outside the circle? This circle is called a
circumscribed circle. Copy the circle and triangle to your word document.
X. Relationships between different lines
1. Start with a new sketchpad.
2. Create and label a triangle.
3. From vertex A, create the altitude, median, and angle bisector.
5-a. Moving vertices can you create a triangle that has all three on the same
line? If so copy it to your word document.
5-b. What kind of triangle was formed? How can you check?
5-c. Can you get only two lines to match up?
4. From vertex B, create the altitude, median, and angle bisector. Mark the
orthocenter “D”, the centroid “E”, and the incenter “F”. Hide all the lines when
done.
5. Construct two perpendicular bisectors and label the circumcenter “G”. Hide the
two lines when done.
6. Move the triangle around and take note of changes of the 4 points.
5-d. What characteristics of the four points do you notice?
5-e. Can you get all 4 to fit on top of each other? If so copy the triangle to your
word document.
5-f. What kind of triangle was formed in the last step? How can you check?
5-g. Are the four points collinear? If you move the triangle around do they stay
collinear?
5-h. Are any three of the points collinear? If so, which three? Select the three
points and construct a line. Are they always collinear when you move the triangle
around? This line is called Euler’s Line. Copy your triangle with Euler’s line to
your word document.