Name: _________________________________ Period: ________ 5.1 Isosceles & Equilateral Triangles An altitude is a perpendicular segment from a vertex to the line containing the opposite side.
1. Prove: the altitude to the base of an isosceles triangle bisects the base.
2. An obelisk is a tall, thin, four sided monument that tapers to a pyramidal top. The Washington
Monument on the National Mall in Washington D.C. is an obelisk. Each face of the pyramidal top is an
isosceles triangle. The height of each triangle is 55.5 feet, and the base of the triangle measures 34.4
feet. Find the length, to the tenth of a foot, of one of the two equal legs of the triangle.
3. With your compass, carefully construct two circles- one with A as a center and AB as the radius, the
other with B as the center and BA as the radius. Label one of their intersections as point C. Use your
straight edge to construct ΔABC.
What kind of triangle is ΔABC? Write a paragraph proof.
Find each value.
6. 7.
Name: _____________________________ Period: ________ 5.2 Bisectors and circumcenters
1. Create the perpendicular bisector of , and create a point on the bisector. How far is the point P from A, and how
far is the point P from B?
2. Create the perpendicular bisectors of the triangles and label the circumcenter as point X. How far is point X from the
vertices of the triangle? Measure them and show you’re correct.
3. Create all 3 bisectors of the triangle and show that they meet at a single point. Then circumscribe the triangle.
4. A group of astronomy students are each
independently working on a project at the
University of Arizona. Jim is at the college of
optical sciences, Claire is at the Steward
observatory, and Carl is located at the University
of Arizona Library.
They all plan to meet and eat lunch on a warm
sunny day, but they all agree that they should all
travel the same distance to meet each other.
Determine the location where they should meet
for lunch.
treasure somewhere on the main island.
Every day they will give a clue as to how to
find their hidden treasure.
The first day the clue is: The treasure is not
near the coast.
treasure is located the same distance from
Mauna Kea, as it is from Mauna Loa.
The next day they give a clue that the
treasure is 28.5 km away from the town of
Mountain View.
Determine the location of the treasure.
Name: _______________________________ Period______ 5.3 Incenter and ∠ bisectors
1) Bisect ∠ angle with ray , show your construction marks.
a) Label point C on the angle
bisector
d) Measure
e) What do you notice?
2) Measure ∠D
on each ray of ∠D.
b) Label these points O and G
c) Create the perpendiculars from
each ray of ∠D through O and G.
d) Label the intersection of the
perpendiculars as point T
F) Measure the angles that are created ∠ODT and ∠GDT.
G) What do you notice?
Incenter
3) Find the angle bisector of each angle of the triangle. Show your work. The place the angle
bisectors intersect is the “incenter” and it is always INSIDE the triangle.
2) You should be able to use the incenter of the triangle to inscribe a circle inside the triangle
(this means the circle is inside of the triangle, the center of the circle is the incenter of the
triangle, and the edge of the circle should just touch each side of the triangle). The incenter is
equal distance to each side of the triangle. Draw each inscribed circle.
3) Legend has it that a treasure ship sank equidistant from the routes that create the Bermuda
Triangle. Use the map below, show all construction marks, and locate where the sunken treasure
lies.
Name: ____________________________Period: ________ 5.4 Medians and Centroids
Median of a triangle is a segment whose endpoints are a vertex of a triangle and the midpoint of
the opposite side. A
B
C
CENTROID : the point of concurrency of the medians of the triangle
2) Find the median of each side of the triangle. Label the centroid as point P. Show your work.
B
A
E
C
F
3) True or False: The point at which a triangular table could balance on
one leg is the same distance from each side of the table.
Name: ____________________________Period: ________ 5.4 Medians and Centroids
P
I
1) Construct a large triangle ΔRST, use compass & straight edge
2) Construct the Circumcenter, label it A
3) Construct the Incenter, label it B
4) Construct the Centroid, label it C
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Worksheet by Kuta Software LLC
Geometry
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-1-
In each triangle, M, N, and P are the midpoints of the sides. Name a segment parallel to the one given.
1) M
5) Find PQ
Worksheet by Kuta Software LLC -2-
7) Find KL
13) Find LN
Name: ___________________________ Period: ______ 5.7 Triangle Properties Quiz Review
1. Match the statements in the table with the words/phrases contained in the word bank
Circumcenter Incenter Centroid
Type of circle
3x + 15
BA=24
∠ = (2 + 12)°
B C
3. Find the value of x 4. Find the value of x
Word Bank:
Perpendicular Bisectors Equidistant from Vertices Medians No circle
Equidistant from Sides
=
=
8. Oscar wants to open a restaurant, but he is concerned with the competition he faces from 3 competitors. If Oscar wants
to build his restaurant an equal distance away from his 3 competitors. Find the location which he should establish his
business. Show all construction marks. Explain your reasoning for why you constructed that location.
9. The safety department at a regional airport has decided to building to
house an emergency response team who will quickly get to the runway to
deal with a crashed aircraft. The inspector decides the building should be
the same distance from all the runways so that emergency vehicles can
quickly get to any of the three runways no matter where a plane may crash.
Determine the location of where to build the building and where to lay
pavement for the vehicles to get to each runway. Explain your reasoning
for why you constructed that location.
Use the square on the left to draw a shape similar to
the image (right). The goal is to produce 4 right
triangles with sides a & b with a hypotenuse c. Sides a
and b should be different lengths.
Do this by marking off equal distances on every side
of the square so that a square with side lengths of c are
formed inside.
Transfer the exact same drawing to the square on the left below. Once
you have drawn it for a second time, please cut out this top portion
and cut out your 4 right triangles and the square.
Name: __________________________ Period: _______ 5.9 Proving The Pythagorean Theorem
Redraw the triangle from above here. Assemble the cut out pieces here.
Using the 4 cut out triangle pieces, assemble them in the square on the right so that the larger square on
the right has 2 squares of different sizes in it with side lengths of a & b.
1. Once you have placed the pieces in the square on the right and the one of the left, explain why 2 + 2
must be equal to 2 in terms of the area they form. Be very specific. You are writing a paragraph proof.
2. Algebraic proof of the Pythagorean Theorem:
a) Use algebra to describe the area of the square on the right using the side
length:
b) Describe the area of the square in terms of all the 5 pieces that make up
the square. (There are 4 congruent triangles and 1 square)
c) What should be true about the area from part 2a, and from part 2b?
d) Use an equation to relating what you said is true in part 2c to show that 2 + 2 = 2
3. Garfield’s Proof: President Garfield proved the Pythagorean Theorem in a very similar manner to what
we did in problem 2, but he used this shape on the right.
Follow a similar process to what we did in problem 2
Name:_____________________________ Period: ________ 5.10 Pythagorean Theorem
You and your partners are the lead project designers for a large company. You are developing new technology that may
allow drones to deliver packages. You are currently testing the software on the drones, and you are verifying that they can
make several deliveries in one trip.
Part a: Use the distances of each street intersection to determine the total length of the flight path your drone will take
(marked with the dotted line). The drone starts at the lower left of the map. The map is on the reverse. Round to two
decimal places. Please label the side lengths of each right triangle.
Part b: Your drone can fly at 30 feet per second. If it stops at each delivery site (marked with an X) for 2 minutes. How
long (in minutes) will it take for the drone to complete the deliveries AND return home? Show your calculations
Part c: A car (which must travel using the streets) can average about 25 miles per hour on the surface streets of Tucson.
The vehicle will stop at each delivery location for about 2 minutes and 30 seconds. In addition the car will have to wait
about 20 seconds at each intersection due to red lights. Calculate the time it would take a car to make the same delivery
route
The map: All units are in feet and indicate the distance between streets. Assume all street corners form right angles and
run parallel to each other.
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Name: _________________________ Period: ________ 5.12 Special Right Triangles
1) Each triangle above is an isosceles right triangle. Use the Pythagorean Theorem to find the length of
the hypotenuse in simplified radical form.
2) How can you calculate the length of the hypotenuse of an isosceles right triangle if you know the
length of the legs?
3) Each triangle below is an isosceles right triangle. Use the Pythagorean Theorem to find the length of
the legs in simplified radical form.
105
4) If you know the length of the hypotenuse, how would you find the length of the legs?
5) Find the missing side lengths 5√2
18 b 14√2
a c = _______
5) In the space above, use a compass and straight edge to construct a large equilateral triangle ΔABC.
What are the measures of each angle ∠A, ∠B and ∠C? _____________________
6) Construct the perpendicular bisector of through point C, label the point D where the perpendicular
bisector intersects . What are the measure of ∠ADC and ∠BDC? ___________________
What is the measurement of ∠ACD and ∠BCD? _____________________ How do you know? _______
7) Is ΔADC ≅ΔBDC? _________________ If so, by which triangle congruence? _____________
8) Is ≅ ? How do you know? ______________________________________
9) How do compare? __________ Will that always be true? ______
10) To the right are two perpendicular lines. Place point M on the horizontal line. Measure a 60 0 angle
from M to the vertical line (extend if needed). Label that point P. You just made a 30-60-90 triangle.
In cm measure ______ and ______ ______
How does the shorter side compare to the longer side of the triangle?
11) Use the Pythagorean Theorem to find the hypotenuse
12) Using the ratio of a 30-60-90 triangle, find the missing side lengths.
12
Worksheet by Kuta Software LLC
TVHS Geometry
Name___________________________________
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Find all missing side lengths. Leave your answers as radicals in simplest form. Show your work.
1)
m
2
3
45°
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Worksheet by Kuta Software LLC
-2-
7)
1) In ABC, what point of concurrency is O?
2) O is equal distance from what part of the
triangle?
3) How is it created?
4) Point G is equal distance from what parts of
the triangle?
6) How do you construct the point of
concurrency in question 5?
8) Describe .
1.3 (diagram above)
11) PQ is the perpendicular bisector or ST. Find
the values of m and n
12) are perpendicular bisectors
PQR. Find
15) is midsegment of RST. What is length
of ?
16) Can 4, 7, 10 be sides of a triangle?
17) Simplify −3√243
19) Solve for x
20) Solve for y.
24) Solve for x
25) Solve for x
5.15 Geogebra Construction Lab
Your work will be submitted to your teacher in a word document or a google doc via
email. You will copy and paste the images of your work from each activity into that file
and then send/share it to your teacher. For the subject line use “Geogebra Lab”
Geogebra Link: www.geogebra.org
Activity 1: To construct an Equilateral Triangle
1. Draw two points A and B using the New Point tool .
2. Draw the line segment AB using the Segment between Two Points
tool .
3. From A draw a circle through B using the Circle with Center through a Point tool .
4. From B draw a circle through A using the Circle with Center through a Point tool
5. Find one intersection C of the two circles using the Intersect Two Objects tool .
6. Draw the line segments AC and BC using the Segment between Two Points tool .
7. Hide the circles by right clicking them and deselecting Show Object.
Activity 2: To construct an isosceles triangle
1. Select the tool (Circle with Center through a Point) and construct a circle center A through point B. If
the labels are not showing, right click, select Properties and with the Basic tab open, click on Show Label.
2. Select the New Point tool and construct any point C on the circumference of circle c.
3. Select the Segment between Two Points tool and construct
[AC]
4. Construct [BC].
5. Right click on one side of the triangle, select Properties, and with the
Basic tab open, click on the drop down arrow beside the Show Label box.
Select Name and Value to show the name and length of this side of the
triangle. Repeat for the other triangle sides.
6. Drag each vertex of triangle ABC and note the length of its sides.
7. Hide the circle, by right clicking on it and clicking on Show Object.
8. Measure the 3 angles in the triangle using the tool. Drag any of
the vertices of the triangle ABC and observe how the angle measures change.
Activity 3: Constructing Medians and constructing the Centroid of a triangle
(A median is a line segment connecting any vertex of a triangle to the midpoint of the opposite side)
1. Click on File and select New Window.
2. Draw a triangle using as above.
3. Using the ,and tools, construct the
medians of each side of the triangle.
4. Construct the intersection of the medians by
selecting the tool.
5. Drag any of the vertices of the triangle and note
that the 3 medians remain concurrent, at the
CENTROID.
circumcircle of a triangle
(A median is a perpendicular bisector of a line segment)
1. Click on File, New Window, and draw a triangle using as above.
2. Select i.e. Midpoint or Center tool and selecting each side of the triangle in turn, construct the
midpoints of each side.
3. Using the Perpendicular Line Bisector tool, select each side to construct perpendicular bisectors
(medians) of each side.
4. Select the Intersect Two Objects tool and then 2 of the medians to construct the circumcenter.
5. The equations of the 3 medians are shown in the Algebra window.
6. Hide the medians by right clicking on each one and clicking on Show Object. Drag the vertices to see
the circumcenter change position.
7. Click on the Circle through a Point tool , then the circumcenter (point of intersection of the
medians) and one of the vertices of the triangle
and construct the circumcircle, which passes
through the 3 vertices.
8. Drag the vertices of the triangle to confirm the
construction.
5.15 Geogebra Construction Lab
Activity 5: Constructing the bisectors of the angles and constructing the
incenter and incircle of a triangle.
1. Construct a triangle ABC in a new window. Select , the Angle Bisector tool. Select the points B, A
and C, in that order, to construct the angular bisector of <BAC. Repeat for the other two angles in the
triangle.
2. Select the tool and 2 of the angle bisectors to construct the incenter.
3. Hide the angle bisector lines.
4. Selecting the tool, draw a perpendicular line from the incenter D, to line AB or any of the 3 sides
of the triangle. With the tool selected construct the intersection E of side AB and this perpendicular
line.
5. Hide the perpendicular line. Select , and with D as center and E
as the point on the circle, construct the incircle.
6. Drag the vertices to confirm the construction.
Activity 6: Find the measurements of the interior angles of a polygon.
1. How does the method of constructing ABCD in steps 1 and 2 guarantee a quadrilateral that is a parallelogram?
2. What are two conditions that must be met for a quadrilateral to be a rectangle? Write a theorem that states the theorem.
5.1 Isosceles and Equilateral
5.3 Incenter and Angle Bisectors
5.4 Centroid and Median
5.12 Special Right Triangles
5.14 Review for Triangle Properties Test
5.15 Geogebra Lab