Download Activity 1a

Computer Number ___
Names _____________________
Date _____________________
Activity 1a
1. Geometer’s Sketchpad has a set of six tools at the left
edge.
Choose the Straightedge Tool:
Click in two places
on the screen and Geometer’s Sketchpad will produce a
line segment connecting two points where you clicked. The
segment will be highlighted.
2. Click on one of the endpoints of the segment
and somewhere else not on the segment.
Geometer’s Sketchpad will produce another line
segment connecting the two points where you
clicked. The new segment will be highlighted and
the old one will no longer be highlighted.
3. Choose the Text Tool. Click on one of the two endpoints
that do not connect the two segments. Geometer’s Sketchpad
will mark that point A. Click on the endpoint that connects
the two segments. Geometer’s Sketchpad will mark that
point B. Click on the remaining endpoint. Geometer’s
Sketchpad will mark that point C.
4. Choose the Selection Arrow Tool:
Click on the
segment that is not highlighted. Now both segments will be
highlighted. This shows that both are selected.
Click somewhere not on the segments. Now nothing will be
highlighted showing that nothing is selected.
Click on the three points A B and C (in that order). They will all be highlighted.
Computer Number ___
Names _____________________
Date _____________________
5. At the top of the Geometer’s Sketchpad window
are nine menus. Click on the Construct menu and
choose Angle Bisector.
Click and drag point A. Observe what happens.
Question 1a-1
What does the Angle Bisector construction do?
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6. The Angle Bisector construction only works if three points are selected and the second
one is the vertex of the angle. In the Construct menu the constructions that are printed in
grey require other objects to be selected.
Select one segment. Observe which constructions listed below are possible. Try them to
see what they do.
Select two points. Observe which constructions listed below are possible. Try them to
see what they do.
Select a point and a segment. Observe which constructions listed below are possible.
Try them to see what they do.
For each construction, record what objects must be selected and what it does:
Construction
Midpoint
Parallel Line
Perpendicular
Line
Circle by Center
and Point
Circle by Center
and Radius
What objects
must be selected
What it does
Computer Number ___
Names _____________________
Date _____________________
Activity 1b
1. Choose New Sketch from the File Menu.
Notice that the Straightedge Tool has a small triangle in one corner:
.
This means you can click and hold it to reveal tools for drawing rays
and lines.
Choose the Line Tool (with two arrows). Click in two places and
Geometer’s Sketchpad will draw a line through those two points.
2. Use the Point Tool
to create a point that is not on the line. Select the line and the
point and use the Parallel Line construction to construct a new line parallel to the first
line and passing through the point. Use the Line Tool again to create a new line that
intersects the two parallel lines.
3. Choose the Point Tool and create points in the positions indicated in the diagram
above. (Notice that Geometer’s Sketchpad indicates when a point will be created on
another object by changing the colour of the object). Choose the Text Tool and click on
the points in the order shown above, to label them A, B, C, etc.
4. Select Points A, B and C (in that order). Click on the
Measure menu and choose Angle. Geometer’s Sketchpad
will display the measure of angle ABC. Measure the other
angles and fill in the first row of the table below. Click and
drag a point and observe how the angle measures change. Fill
in the second row.
Angle
ABC ABD CBE DBE BEF BEH FEG HEG
Measure before dragging
Measure after dragging
Computer Number ___
Names _____________________
Date _____________________
Questions
1b-1 Which angles have the same measure as angle ABC?_____________________
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1b-2 The relationship between angles ABD and BEH is sometimes described by
saying it makes a letter F.
What other pairs of angles
make an F? (You may need
to imagine an F upside
down or in a mirror)
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1b-3 The relationship
between angles DBE and BEF
is sometimes described by
saying it makes a letter Z.
What other pairs of angles
make a Z? (You may need to
imagine a Z upside down or
in a mirror)
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1b-4 The relationship between angles ABC and DBE is sometimes described by
saying it makes a letter X. What other pairs of angles make an X?
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1b-5 Which angles’ measures add up to 180°? ______________________________
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1b-6 How would you describe the relationship between the angles whose measures
add up to 180°?
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Computer Number ___
Names _____________________
Date _____________________
Activity 2a
1. Open the file Six_parameters.gsp. You will see six
measurements, three lengths (5 cm, 6 cm and 7 cm)
and three angle measure (50°, 60° and 70°). These
measurements are called parameters. Double click
one of the parameters. A window will open in which
you can change the value of the parameter by typing
a new number and clicking OK. If the parameters are
not 5 cm, 6 cm and 7 cm and 50°, 60° and 70°,
change the ones that are wrong.
2. The Custom Tool includes new tools
that have been added to Geometer’s
Sketchpad.
This document includes two new tools: Angle from
Parameter and Segment from Parameter.
3. Choose Segment from Parameter.
Click on a blank area to create a new point. This point will be one endpoint of the
segment.
Click on a parameter that gives a length. Geometer’s Sketchpad will create a segment
with that length. Drag the endpoints of the segment. Notice that the length always stays
the same.
Choose the Selection Arrow Tool
and double click the parameter you used for the
length of the segment. Change the value of the parameter. Observe what happens to the
segment. Change the value of the parameter back to its original value.
4. Choose the Custom Tool Segment from Parameter again. Click on one endpoint of
the segment you created. Then click on a different length parameter than you used the
first time. Geometer’s Sketchpad will create a new segment connected to the first one.
Create a third segment connected to the free endpoint of one of the segments.
Questions
2a-1 Can you drag the three segments so they form a triangle? _________________
2a-2 If so, what are the angle measures of the triangle? _______________________
2a-3 If not, why can’t you form a triangle? _________________________________
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Computer Number ___
Names _____________________
Date _____________________
5. If you could not create a triangle with the three given length parameters, change the
parameters so you can form a triangle. Record your new parameters here: ____________
Drag the three segments and try to form two different triangles with sides of the same
three lengths.
Questions
2a-4 Can you drag the three segments so they form two different triangles? _______
2a-5 If so, what are the angle measures of the two triangles? ____________________
2a-6 If not, why can’t you form two different triangles? ________________________
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2a-7 Can you always form at least one triangle with sides measuring three given
lengths?
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2a-8 If so, can you explain why this is possible? _______________________________
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2a-9 If not, can you explain when this is impossible? ___________________________
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Computer Number ___
Names _____________________
Date _____________________
Activity 2b
1. If you have a file open, choose Save As… from the File menu. Save your file, calling it
“Activity 2a”. Open the file Six_parameters.gsp again. You will see the six parameters. If
the angle measure parameters are not 50°, 60° and 70°, change the ones that are wrong.
2. Choose the Custom Tool Angle from Parameter.
Click on a blank area to create a new point. This point will be the vertex of the angle.
Click somewhere else to create another point. This point will be on one arm of the angle.
Click on a parameter that gives an angle measure. Geometer’s Sketchpad will create a
angle with that angle measure. Drag the points of the angle. Notice that the angle
measure always stays the same.
Change the value of the parameter you used for the measure of the angle. Observe what
happens to the angle. Change the value of the parameter back to its original value.
4. Choose the Custom Tool Angle from Parameter again. Click on the newest point of
the angle you created. Then click on the vertex. The click on a different angle measure
parameter than you used the first time. Geometer’s Sketchpad will create a new angle
connected to the first one. Create a third angle connected to the new angle in the same
way.
Questions
2b-1 Can you drag the three angles so they form a triangle? ____________________
2b-2 If so, what are the lengths of the sides of the triangle? _____________________
2b-3 If not, why can’t you form a triangle? ___________________________________
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Computer Number ___
Names _____________________
Date _____________________
5. If you could not create a triangle with the three given angle measure parameters,
change the parameters so you can form a triangle. Record your new parameters here:
___________________________________________________________________
Drag the three angles and try to form two different triangles with angle of the same three
measures.
Questions
2b-4 Can you drag the three angles so they form two different triangles? _________
2b-5 If so, what are the side lengths of the two triangles? _______________________
2b-6 If not, why can’t you form two different triangles? ________________________
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2b-7 Can you always form at least one triangle with angles of three given measures?
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2b-8 If so, can you explain why this is possible? _______________________________
________________________________________________________________________
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2b-9 If not, can you explain when this is impossible? ___________________________
________________________________________________________________________
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________________________________________________________________________
Computer Number ___
Names _____________________
Date _____________________
Activity 2c
1. Choose Save As… from the File menu. Save your file, calling it “Activity 2b”. Open
the file Six_parameters.gsp again. You will see the six parameters. If the angle measure
parameters are not 50°, 60° and 70°, change the ones that are wrong.
2. In this activity you are going to explore whether it is possible to construct many
triangles, a unique triangle or no triangles at all given two angle measures and one side
length. There are two cases: the given side can be between the two angles, or it can be
one of the two sides not between the two angles. The abbreviations ASA and AAS stand
for these two possibilities.
ASA
AAS
3. Choose the Custom Tool Segment from Parameter. Create a segment using one of
the three length parameters. Choose the Custom Tool Angle from Parameter. Create an
angle by clicking on the two endpoints of the segment and one of the angle measure
parameters (Recall that the first point you click on with the Angle from Parameter tool is
the vertex of the angle). Create another angle by clicking on the two endpoints of the
segment in the opposite order and a different angle measure parameter.
You will notice that the angle goes the wrong way to create a triangle. Instead it makes a
Z shape. This is because the Angle from Parameter tool always makes the angle going
counter clockwise from the segment. To get the angle to go the direction you want,
change the parameter to its negative.
Before: 60° angle
After: -60° angle
4. Drag the points and try to make a different triangle with the same two angle measures
and side length.
Computer Number ___
Names _____________________
Date _____________________
Question 2c-1
Based on your investigations do you think it is possible to construct many triangles,
a unique triangle or no triangles at all, given two angle measures and one side
length, when the given side is between the two angles (ASA)?
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5. Choose the Custom Tool Segment from Parameter. Create a new segment using one
of the three length parameters. Choose the Custom Tool Angle from Parameter. Create
an angle by clicking on the two endpoints of the segment and one of the angle measure
parameters. Create another angle by clicking on the two endpoints of the new segment
created by the Angle from Parameter tool. Click on the free endpoint first so that the
angle has that point as its vertex.
6. Drag the points and try to make a different triangle with the same two angle measures
and side length.
Question 2c-2
Based on your investigations do you think it is possible to construct many triangles,
a unique triangle or no triangles at all, given two angle measures and one side
length, when the given side is not between the two angles (AAS)?
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Computer Number ___
Names _____________________
Date _____________________
Activity 2d
1. Choose Save As… from the File menu. Save your file, calling it “Activity 2c”. Open
the file Six_parameters.gsp again. You will see the six parameters. If the angle measure
parameters are not 50°, 60° and 70°, change the ones that are wrong.
2. In this activity you are going to explore whether it is possible to construct many
triangles, a unique triangle or no triangles at all given two side lengths and one angle
measure. There are two cases: the given angle can be between the two sides, or it can be
one of the two angles not between the two sides. The abbreviations SAS and SSA stand
for these two possibilities.
Given side 1
Given ang le
Given side 1
Given ang le
Given side 2
Given side 2
SAS
SSA
3. Choose the Custom Tool Segment from Parameter. Create a segment using one of
the three length parameters. Choose the Custom Tool Angle from Parameter. Create an
angle by clicking on the two endpoints of the segment and one of the angle measure
parameters. You now have an angle with one arm of a fixed length and the other one of
variable length. To create the SAS configuration it is necessary to replace the variable
arm with one of fixed length. To do this, select the vertex of the angle and a length
parameter. Choose the Circle by Center+Radius construction from the construct menu.
Computer Number ___
Names _____________________
Date _____________________
Geometer’s Sketchpad will create a circle with the given
radius. Create a point where the circle crosses the variable
length arm of the angle (If the variable length arm is not long
enough to intersect with the circle, drag it to make it longer).
Create a segment joining the intersection point with the
vertex of the angle. You won’t be able to see it because it
overlaps with the variable length arm.
Select the circle and
the variable length
arm of the angle.
Choose Hide Path
Objects from the
Display menu.
Geometer’s
Sketchpad will hide
the circle and the
variable length arm of
the angle, leaving an
angle with two fixed
length arms.
4. Drag the points and try to make a different triangle with the same two side lengths and
angle measure.
Question 2d-1
Based on your investigations do you think it is possible to construct many triangles,
a unique triangle or no triangles at all, given two side lengths and one angle
measure, when the given angle is between the two sides (SAS)?
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Computer Number ___
Names _____________________
Date _____________________
5. Summarise your findings so far in this table:
Case
No triangle
A unique triangle
possible
possible
SSS (Three side
lengths given)
AAA (Three angle
measures given)
ASA
AAS
SAS
Many triangles
possible
Question 2d-2
Based on your findings so far, what would you predict for the SSA case (two side
lengths and one angle measure given, with the given angle not between the two
sides): many triangles, a unique triangle or no triangles at all? Why?
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6. Use the Custom Tool Segment from Parameter to create a segment 5 cm long. Use
Angle from Parameter to create a 50° angle at one end of the segment. Use Segment
from Parameter to create a segment 4 cm long at the other end of the segment. Drag the
points and try to make different triangles with the same two side lengths and angle
measure.
Question 2d-3
Based on your investigations do you think it is possible to construct many triangles,
a unique triangle or no triangles at all, given two side lengths and one angle
measure, when the given angle is not between the two sides (SSA)?
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
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Computer Number ___
Names _____________________
Date _____________________
Activity 3a
For this activity you will need to recall these definitions:
 A quadrilateral is a polygon with four sides.
 A square is a quadrilateral with four sides of equal length and four 90° angles.
 A rectangle is a quadrilateral with four 90° angles.
 A rhombus a quadrilateral with four sides of equal length.
 A parallelogram is a quadrilateral with two pairs of parallel sides.
1. To construct a square you
can use information from the
definition. Begin by
constructing a segment.
To create 90° angles at each
endpoint use the
Perpendicular Line
construction. Select the
segment and one endpoint
and choose Perpendicular
Line from the Construct
menu. Do the same at the
other endpoint of the
segment.
To create sides of length
equal to the first segment use
the Circle by
Center+Radius
construction. Select the
segment and one endpoint
and choose Circle by
Center+Radius from the
Construct menu. Then create
a point where the circle and
the perpendicular line
intersect. Do the same at the
other endpoint of the
segment.
Hide the circles and lines
and connect the points with
segments. Drag the square to
make sure it remains a
square. (This way of
checking a construction is
called the “Drag Test”.)
Computer Number ___
Names _____________________
Date _____________________
2. Construct the two diagonals of the square by making segments joining the opposite
corners. Drag the square and observe the diagonals.
Question 3a-1
What have you observed about the diagonals of squares? _______________________
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3. You have probably observed many things about the diagonals, including that they are
both the same length, that they meet at a 90° angle, and that the point where they meet is
the midpoint of both diagonals. Dragging the square, or perhaps making some
measurements, is probably enough to convince you that these three properties are true for
the diagonals of all squares. But knowing that something is true is not the same as
knowing why something is true.
Read over this explanation of the three properties of diagonals of squares:
Diagonal AC is the same length as diagonal
BD because they are corresponding parts of
congruent triangles. Triangle ABC is
congruent to triangle BAD by side-angle-side,
because segments AB and BA have the same
length (they are the same segment), segments
ABC and AD have the same length (all the
sides of a square have equal length), and angle
ABC has the same measure as angle BAD (all
the angles of a square measure 90°).
Angle AEB measures 90°. This is because
angles BAC and DAC have the same measure
(they are corresponding parts of congruent
triangles, see above), and together they make
up angle BAD which measures 90°; therefore
they must each measure 45°. Similarly all the
angles made by the diagonals with the sides of
the square must measure 45°. So in triangle
ABE, angles BAE and ABE measure 45°. As
the angles of a triangle add up to 180°, the
third angle, AEB, must measure 90°.
A
B
A
B
D
C
D
C
A
B
A
B
E
D
E
C
D
C
Computer Number ___
Diagonals AC and BD meet at their
midpoints. This is because triangles ABE,
BCE, CDE, and DAE are all congruent by
angle-side-angle (we know from above that all
the angles made by the diagonals with the
sides of the square measure 45°, and all the
sides of a square have the same length).
Because they are corresponding parts of
congruent triangles, AE must be the same
length as CE, so E is the midpoint of AC, and
BE must be the same length as DE, so E is the
midpoint of BD.
Names _____________________
Date _____________________
A
B
A
E
D
B
E
C
D
C
Record here any questions or comments you have about this explanation:
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4. You now know that:
IF a quadrilateral is a square THEN the diagonals are of equal length, meet at a
90° angle and meet at their midpoints.
But it is not always possible to turn such a statement around the other way. It is true that:
IF a person lives in Antigonish THEN that person lives in Nova Scotia,
but it is not true that:
IF a person lives in Nova Scotia THEN that person lives in Antigonish.
Now you will investigate whether:
IF a quadrilateral has diagonals that are of equal length, meet at a 90° angle and
meet at their midpoints THEN the quadrilateral is a square.
To do this, first construct two segments that are of equal length, meet
at a 90° angle and meet at their midpoints. It is easiest to begin with the
90° angle and the midpoint. Create a segment. Construct its midpoint.
Use the Perpendicular Line construction to construct a perpendicular
line through the midpoint.
Computer Number ___
Names _____________________
Date _____________________
Now you need to create a segment on this line that is the same length
as the original segment and which has the same midpoint. Construct a
circle with the original segment as its diameter using the Circle by
Center+Radius construction. Where the circle intersects with the
perpendicular line make two points, which are the endpoints of the
second diagonal. Hide the circle and the perpendicular line.
Join the four endpoints of the two diagonals to make a quadrilateral.
Drag your figure and observe the quadrilateral. Does it seem to be a
square?
Provide an explanation for why IF a quadrilateral has diagonals that are of equal
length, meet at a 90° angle and meet at their midpoints THEN the quadrilateral is a
square. Use the back of the page if necessary.
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Computer Number ___
Names _____________________
Date _____________________
Activity 3b
1. To describe a square one might want to include everything one knows about squares:
They have four equal sides, four 90° angles, the diagonals are the same length, and meet
at their midpoints at a 90° angle, etc. But for a mathematical definition, it is best to
include as few properties as possible. As you have found out above if you know that a
quadrilateral has four equal sides and four 90° angles then the diagonal properties must
also be present. There is not need to state them in the definition. In fact, it could be
argued that the definition of a square given above “A square is a quadrilateral with four
sides of equal length and four 90° angles” includes too much information. As you know
that all quadrilaterals have four angles that add up to 360°, all you need to say is that “A
square is a quadrilateral with four sides of equal length and four equal angles.” You can
then deduce that the angles must measure 90°.
In Activity 3a you found that if a quadrilateral has diagonals that are of equal length,
meet at a 90° angle and meet at their midpoints then the quadrilateral is a square. It is
possible that you don’t need to know all three diagonal properties to establish that a
quadrilateral is a square. In the following you will investigate whether a quadrilateral
with only two of the diagonal properties is a square. There are three cases to explore:
 diagonals that are of equal length and meet at their midpoints
 diagonals that meet at a 90° angle and meet at their midpoints
 diagonals that are of equal length and meet at a 90° angle
2. Construct two diagonals that are of equal length and meet at their midpoints. To do
this, create a segment, construct its midpoint, and construct a circle centered on the
midpoint with the segment as a diameter. Any other diameter of the circle will be the
same length as the original segment. Hide any construction elements you don’t need
anymore. Join the four endpoints of the two diagonals to make a quadrilateral. Drag
your figure and observe the quadrilateral. Does it seem to be a square?
Record your observations here:
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________________________________________________________________________
________________________________________________________________________
Computer Number ___
Names _____________________
Date _____________________
3. Construct two diagonals that meet at a 90° angle and meet at their midpoints. To do
this, create a segment, construct its midpoint, and construct a perpendicular line through
the midpoint. Create a point on the perpendicular line and a circle with its center at the
midpoint and passing through the new point. Create a second point where the circle also
meets the perpendicular line. Connect those points with a segment, which is the second
diagonal you want. Hide any construction elements you don’t need anymore. Join the
four endpoints of the two diagonals to make a quadrilateral. Drag your figure and
observe the quadrilateral. Does it seem to be a square?
Record your observations here:
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4. Construct two diagonals that are of equal length and meet at a 90° angle. To do this,
create a segment, construct its midpoint, and construct a second point anywhere on the
segment. Construct a perpendicular line through the second point. Create a new point on
the perpendicular line. Measure the distance from the midpoint to one endpoint. Create a
circle with its center at new point on the perpendicular line and using the distance you
measured as the radius. Create two points where the circle meets the perpendicular line.
Connect those points with a segment, which is the second diagonal you want. Hide any
construction elements you don’t need anymore. Join the four endpoints of the two
diagonals to make a quadrilateral. Drag your figure and observe the quadrilateral. Does
it seem to be a square?
Record your observations here:
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
Computer Number ___
Names _____________________
Date _____________________
________________________________________________________________________
5. Construct a parallelogram, by beginning with two segments joined at one endpoint, and
constructing lines parallel to each segment through the free endpoint of the other
segment.
What properties do the diagonals of a parallelogram have?
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Construct two diagonals with those properties.
Is the quadrilateral formed by the four endpoints of the diagonals necessarily a
parallelogram. Why?
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Computer Number ___
Names _____________________
Date _____________________
Activity 4a
Create a triangle. Using the Midpoint and Perpendicular Line constructions, construct
the perpendicular bisectors of two sides of the triangle. Create a point at the intersection
of the two perpendicular bisectors. Measure the distance from the intersection point to the
three points of the triangle. Drag the triangle’s points and observe what happens to the
distances. Construct a circle with its center at the intersection point of the perpendicular
bisectors and passing through one point of the circle. Drag the triangle’s points and
observe what happens to the circle.
Record your observations here:
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Explain the properties of the distances and the circle that you observed by referring
to congruent triangles (Hint: you may want to connect the intersection point to the
points of the triangle with segments.)
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Computer Number ___
Names _____________________
Date _____________________
Activity 4b
Create a triangle. Using the Angle Bisector construction, construct the angle bisectors of
two angles of the triangle. Create a point at the intersection of the two angle bisectors.
Measure the distance from the intersection point to the three sides of the triangle. Drag
the triangle’s points and observe what happens to the distances. Construct a circle with its
center at the intersection point of the angle bisectors and with its radius equal to one of
the distances you measured. Drag the triangle’s points and observe what happens to the
circle.
Record your observations here:
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Explain the properties of the distances and the circle that you observed by referring
to congruent triangles (Hint: you may want to use segments to connect the intersection
point to the points on the sides where the circle touches.)
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