Name______________________________________________Period______Date due: Friday, November 20, 2015

TOPIC 5 LESSON 2: MIDSEGMENT OF A TRIANGLE

A midsegment of a triangle is the segment that connects the midpoints of two sides of the triangle. A triangle has three midsegments, but we will just investigate one midsegment.

Scenario: A golfer has hit his ball to point D. He needs to figure out the distance across the pond to point E. How can he do this?

Pond ongolf course

mADE =mABC =

=DEBC

DE =BC =

F

E

D

A

B

C

1. Construct triangle ABC.2. Find the midpoint of each side.3. Construct one midsegment from D to E.4. Measure the midsegment. DE = ________5. Measure the third side of the triangle that

doesn’t contain the endpoints of the midsegment. BC = _______

6. Use the calculator under Number to compute

the ratio =

7. Write a conjecture concerning the length of the midsegment compared to the length of the third side of the triangle. Remember to be very specific when writing conjectures!

8. Measure ADE = _____ and ABC = ________.9. Are these two angle measures equal or not? ____10. Write a conjecture concerning the relationship

between the midsegment and the third side of a triangle.

11. Get my initials before you erase your screen!

THREE MIDSEGMENTS OF A TRIANGLE

Each person in your group will need ½ piece of patty paper. This will not be used until step #4!

1. Draw a triangle (any type) on the right side of this page. Have one person in your group draw an acute triangle, another person draw an obtuse triangle, another person draw a right triangle, and another person draw an equiangular (omit this type if your group has only 3 people).

2. Measure each side of the triangle using your ruler and locate the midpoint.

3. Draw the three midsegments. You should now have 4 small triangles.

4. Place the ½ piece of patty paper over the triangle below and copy (trace) one of the four triangles.

5. Compare all four triangles by sliding the copy of one small triangle over the other three triangles. Compare your results with the results of your group.

6. Write a conjecture below about your discovery.7. Get my initials after showing me your patty

paper and writing your conjecture.

CONJECTURE:

TOPIC 5 LESSON 3: INVESTIGATING DISTANCE ON THE PERPENDICULAR BISECTOR OF A

SEGMENT USING GEOMETER’S SKETCHPAD

Scenario: Abilene and Browning are towns that want to share the cost of an airport. There is a road, Highway 33, that connects Abilene and Browning. There is a road, Highway 57, that connects Denton and Kacey. The mayors of both Abilene and Browning want to build the airport so that it is equidistant from both towns. Where do they build the airport?

Colton

Abilene

Browning

Denton

Kacey

Sketchpad directions:1. Construct , then construct the midpoint C.2. To construct the perpendicular bisector of the

segment, you need to select the midpoint and the segment (not the endpoints of the segment!). Then construct the perpendicular line.

3. Place a point D on the perpendicular bisector.

4. Construct and . Highlight both of these segments and change it to a dashed line under Display, line style, dashed.

5. Measure the length or distances of the segments.DA = ________ DB = ________

6. Is DA = DB? __________

7. Move point D up and down the perpendicular bisector. Does DA = DB? __________

8. Fill in the blanks for this conjecture:Hint: words in the title and the word equidistant should go in this conjecture!

If a _________________ is on the __________

________________ of a _______________, then

it is _________________________ from the

________________________ of the

___________________.

9. Where should the mayors agree to build the airport? Consider whether there's only 1 place, 3 places, or an infinite # of places!

10. Get my initials before you erase your screen!

TOPIC 5 LESSON 3: INVESTIGATING DISTANCES ON AN ANGLE BISECTOR

Scenario: Player A is dribbling the soccer ball towards the goal. Player A can dribble to the left, or right, or straight at the goal. Where should the goalie stand in order to be in the best position to defend the goal?

D

E

F

Player AGoalie

Goal post #1 (point B)

Goal post #2 (point C)

Sketchpad directions:1. Use the ray tool to draw BAC.2. Select points B, A, C (in order) to name the .3. Then construct the angle bisector of BAC.4. Place a point D on the angle bisector.5. DEFINITION: Distance from a point to a line (or segment or ray) is always the length of the perpendicular segment from the point to the line (or segment or ray).6. Construct the perpendicular line from point D to each side of the angle. Select point D and of BAC (NOT the angle bisector), and construct the perpendicular line. In the diagram this is shown as a dashed line. You can change it to a dashed line under Display, line width, dashed.

7. Put point E where the right angle is formed.

8. Repeat with the other side of the angle of BAC and put a point F.

9. Measure DE = ________ and DF = ________.

10. Are these distances equidistant? _____________

11. Move point D up and down the angle bisector.

12. Are these distances still equidistant? __________

13. Fill in the blanks for this conjecture:Hint: words in the title and the word equidistant should go in this conjecture!

If a ___________________ is on the _________

____________ of an _____________, then it is

__________________________ from the

______________ of the ______________.

14. Where does the goalie need to stand? Consider whether there's only 1 place, or an infinite # of places!

15. Get my initials before youerase your screen!

TOPIC 5 LESSON 4: INVESTIGATION OF SPECIAL SEGMENTS OF A TRIANGLE:

PERPENDICULAR BISECTORS OF SIDESIN A TRIANGLE

Scenario: The 3 small country schools in Ashtown, Bucktown, and Carltown want to build one stadium to be used by all three schools that is equidistant from the 3 towns. Where do they build the stadium?

AG = ______BG = ______CG = ______

G

F

E

D

Bucktown

Ashtown

Carltown

1. Construct triangle ABC.2. Construct the perpendicular bisectors of each side

following these steps:a. Construct the midpoint of a side.b. Select the midpoint and that side, and construct

perpendicular line.c. Repeat for a second side.d. Construct the third perpendicular bisector.e. Select two of the perpendicular bisectors and

construct the intersection.3. Two or more lines are concurrent if they intersect in a

single point. Are the 3 perpendicular bisectors concurrent? ________ This point is called the circumcenter.

4. Construct the segment from each vertex to the circumcenter. Make it dashed as shown.

5. Measure AG = _________, BG = _________, and CG = _________.

6. Move one or more of the vertices. Do these distances stay equal to each other? ______

7. Select the circumcenter and any vertex (in that order!), then using Construct menu choose circle by center + point.

8. Does the circle contain vertices A, B, and C of the triangle? ____________ What vocabulary word represents ?

This is called a circumscribed circle.9. Fill in the blanks for this conjecture: Hint: words in the

title and the word equidistant should go in this conjecture!The ____ _____________________ of a triangle are __________________ in a point called the ______________________________. The ____________________________ is _____________________ from the _________________ of the triangle.

10. Where do they build the stadium? Answer with a vocabulary word!

11. Manipulate your construction to form the four different types of triangles and notice the location of the circumcenter. Fill in the chart with the words "inside", "on", or "outside" the triangle.

TYPE OFTRIANGLE

CIRCUMCENTER

ACUTEOBTUSE

RIGHT

EQUIANGULAR

12. Get my initials before you erase your screen!

TOPIC 5 LESSON 4: INVESTIGATION OF SPECIAL SEGMENTS OF A TRIANGLE:

ANGLE BISECTORS OF A TRIANGLE

Scenario: Grandma Dora has a piece of land that has roads from point A to point B to point C. She will let each of her 3 grandchildren Evan, Fred, and Gretchen build a house on the land if and only if they meet 2 requirements: they must build her a house also; and the roads from her house to each of the grandchildren's houses must be equidistant. Find out where to build the 4 houses.

DE =DF =DG =

Gretchen

Fred

Evan

Dora

A

B

C

1. Construct triangle ABC.2. Construct angle bisectors following these steps:

a. Select three vertices to name an angle, such as BAC. Then construct the angle bisector.

b. Repeat with a second angle.c. Put a point where these two angle bisectors

intersect.d. Construct the third angle bisector.

3. Are the angle bisectors concurrent? __________ This point is called the incenter.

4. Construct the perpendicular line (select point D and the side of the triangle) from the incenter to each of the three sides. The diagram shows this as a dashed line (under Display, line width, choose dashed).

5. Put a point where the perpendicular line meets the side of the triangle. Repeat for other 2 sides.

6. Measure the distance from the incenter to each side of the triangle (remember that distance from a point to a line or segment is always perpendicular!).DE = ______ DF = ______ DG = ________

7. Move one of more vertices. Does DE = DF = DG always? ________

8. Select the incenter and either E, F, or G. Construct a circle by center + point. Does the circle touch each side of the triangle? ________ This is called an inscribed circle.

9. Fill in the blanks for this conjecture:The _______ _________________ of a triangle are _____________________ in a point called the ______________________. This point is _________________________ from the ____________ of the triangle.

10. Manipulate your construction to form the four different types of triangles and notice the location of the incenter. Fill in the chart with the words "inside", "on", or "outside" the .

TYPE OFTRIANGLE INCENTER

ACUTEOBTUSERIGHT

EQUIANGULAR11. Get my initials before you erase your screen!

TOPIC 5 LESSON 5: INVESTIGATION OF SPECIAL SEGMENTS OF A TRIANGLE:

MEDIANS OF A TRIANGLEA median of triangle is a segment that connects one vertex with the midpoint of the opposite side. The centroid is the center of gravity of a triangle. Two or more lines are concurrent if they intersect in a single point.

=CGCF

=BGBE

=AGAD

AG =BG =CG =

AD =BE =CF =

G

F E

D

A

BC

1. Draw a triangle ABC.2. Construct 3 midpoints D, E, F.3. Construct 2 medians.4. Put a point G where the two medians intersect.5. Construct the third median.6. Do the medians intersect concurrently or not? _____

This point is called the centroid. Move each vertex of the triangle around. Do the medians stay concurrent? _____

7. Measure each median.

8. Measure the distance from each vertex to the centroid.9. Go to Edit, Preferences, and set the precision for ratio

to hundred thousandths. Use the calculator under Number to compute each ratio for all three medians:

= =

10. Were all of the ratios equal? _____

11. Do these ratios stay equal when you move one or more vertices? _____

12. This ratio is given in decimal form. What fraction is equivalent to this ratio? __________

13. Fill in the blanks for this conjecture:

The _________________ of a triangle are ____________. They intersect in a point called the _______________. The _________________ of a

triangle is located (put a fraction!) of the distance

from each ______________ to the ________________ of the opposite side.

14. Manipulate your construction to form the four different types of triangles and notice the location of the centroid. Fill in the chart with the words "inside", "on", or "outside" the .

TYPE OFTRIANGLE CENTROID

ACUTEOBTUSERIGHTEQUIANGULAR

15. Get my initials before you erase your screen!

INVESTIGATION OF THE CENTER OF GRAVITY OF A TRIANGLE.

1. Cut out the triangles given to your group. Cut carefully on the line segments!!!

2. Try to balance your triangle on your finger or on the tip of your pencil.

3. Mark this center of gravity with your pencil on the triangle.

4. Find the midpoint of each side by measuring each side length in cm. Be as accurate as possible! Repeat for all 3 sides.

5. Connect this midpoint to the opposite vertex using a straight edge. This segment is called a median of a triangle.

6. Are the three medians for your triangle concurrent? ________ This point is called the centroid.

7. Now try to balance your triangle on a fingertip. Where does it balance? On the point that you marked at the beginning, or at the centroid? ______________________

8. Measure the length of one median. Measure the two parts: from the midpoint to the centroid, and from the centroid to the vertex. Answer the following questions with a fraction:a. The distance from the vertex to the centroid is ___________ the length of the median.b. The distance from the midpoint to the centroid is _________ the length of the median.c. What is the ratio of the above answers b:a? ______

For #8-10, use ΔABC with midpoints D, E, F, and centroid G.8. If AD = 24, then AG = _____ and GD = ______.

9. If EG = 7 ½ , then GB = ______ and EB = ______.

10. If CG = 6⅔, then GF = ______ and CF = ______.

11. Get my initials when this page is complete.A

B

F

E

TOPIC 5 LESSON 5: INVESTIGATION OF SPECIAL SEGMENT IN A TRIANGLE:

ORTHOCENTER

Definition: The altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. Every triangle will have three altitudes. An altitude can be inside, outside, or on the triangle.

D

A

B

C

1. Construct triangle ABC.

2. Construct the altitudes to each side following these steps:a. Select a vertex and the side of the triangle

opposite the vertex.

b. Construct the perpendicular line to represent the altitude. Change the color to red.

c. Repeat for a second side.d. Put a point where the two perpendicular

bisectors intersect.e. Repeat for the third side.

3. Are the three altitudes concurrent? ________This point is called the orthocenter.

4. Move one or more of the vertices. Does the orthocenter stay concurrent?________

5. Measure AD = _______, BD = _______, and CD = ______. Are they equidistant?______

6. Manipulate your construction to form the four different types of triangles and notice the location

of the orthocenter. Fill in the chart with the words "inside", "on", or "outside" the triangle.

TYPE OFTRIANGLE

ORTHOCENTER

ACUTEOBTUSERIGHT

EQUIANGULAR

7. The poor orthocenter doesn’t have anything special about it! L L L So there’s no conjecture to write this time!

C

D

8. Get my initials before you erase the sketch!

TOPIC 5 LESSON 5 (without Sketchpad)

The ORTHOCENTER is the concurrent point formed by the altitudes of a triangle. Use a straightedge and a compass to construct the altitudes from all three sides of each different type of triangle. Discover whether the orthocenter is inside the triangle, on the triangle, or outside the triangle.

To construct the altitude, follow these steps. I will use ΔABC for the example.

1. Put the compass point on B and draw an arc that will intersect segment AC twice. Extend segment AC outside the triangle if necessary for this to occur.2. Put the compass point on one of these intersections (point U). Put the pencil on the vertex you are working with (point B). Draw an arc. Repeat using point V.3. Use the straight edge to connect the intersections of these two arcs (points B and Y). This should pass through the vertex of the triangle and be perpendicular to the side of the triangle, thus forming the altitude segment BW.4. Repeat for the other two vertices.5. Use a highlighter and highlight the 3 altitudes.6. Put a point to represent the orthocenter.

W

Y

VU C

B

AX

Use the triangles on the next page and your compass and straightedge to construct the orthocenter. Each person in your group should choose a different triangle. Fill in the chart with the words "inside", "on", or "outside" the triangle.

TYPE OFTRIANGLE

ORTHOCENTER

ACUTEOBTUSERIGHT

EQUIANGULAR

The poor orthocenter doesn’t have anything special about it! L So there’s no

conjecture to write this time!

mBCA = 41mABC = 77mBAC = 62

A

B

C

mEFD = 39mDEF = 114mEDF = 27

D

E

F

mNPO = 28mONP = 62mNOP = 90

N

O P

mJLK = 60mJKL = 60mKJL = 60

L

J K

After you do all 4 of the investigations on circumcenter, incenter, centroid, and orthocenter, see if you can figure out what this acronym ABI PBC AO MC stands for. Then make up a cute saying (example PEMDAS or Please Excuse My Dear Aunt Sally) to help you remember! You can rearrange the grouping of letters if you want!

Your Cute Saying: A previous class of Pre-AP made this cute saying:

A _ _ _ _B _ _ _ _ _ _ _ _I _ _ _ _ _ _ _

P _ _ _ _ _ _ _ _ _ _ _ _B _ _ _ _ _ _ _ _C _ _ _ _ _ _ _ _ _ _ _

A _ _ _ _ _ _ _ _O _ _ _ _ _ _ _ _ _ _

M _ _ _ _ _ _C _ _ _ _ _ _ _TOPIC 5 LESSON 7: INVESTIGATION OF ANGLES AND SIDES OF A TRIANGLE

Draw an acute, right, or obtuse triangle and measure all three sides and all three angles. Do NOT draw an isosceles or equilateral triangle. Remember that it is easier to measure with your protractor if you draw a LARGE triangle!

Then label the largest angle A, the smallest angle C, and the other angle B to form the triangle ABC.

Use LETTERS, not numbers to answer the following questions:

Which is the shortest side? ________Which is the smallest angle? _______

Which is the medium side? ________Which is the medium angle? _______

Which is the longest side? _________Which is the largest angle? ________

Write 3 separate conjectures:1.

2.

3.

TOPIC 5 LESSON 5: INVESTIGATION OF INEQUALITIES IN ONE TRIANGLEDirections: choose any 3 of the random ruler lengths and try to form a triangle.

Record your results and the results of your group below.

YES, these segment lengths form a triangle.

Shortest length

Medium length

Longest length Write a conjecture:

NO, these segment lengths do not form a triangle.

Shortest Medium Longest

length length length

X

Y

Z

How do you find the range of numbers for the third side of a triangle if you are given two sides?