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.
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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.
4. 𝑚∠𝐴 = ______ 5. 𝐶𝐴 = _______
6. 7.
𝑓𝑖𝑛𝑑 𝐷𝐺 = ________ 𝑓𝑖𝑛𝑑 ∠𝑇 = ______
8. 9. 𝑚∠𝐷 = ______
𝑡 = _____
𝑓𝑖𝑛𝑑 ∠𝐴𝐵𝐶 = ______ 10. 11.
𝑓𝑖𝑛𝑑 ∠𝐻 = ______ 𝑓𝑖𝑛𝑑 ∠𝐴𝐶𝐷 = ______
𝑓𝑖𝑛𝑑 ∠𝐴𝐷𝐶 = ______
Name: _____________________________ Period: ________ 5.2 Bisectors and circumcenters
1. Create the perpendicular bisector of 𝐴𝐵, and create a point 𝑃 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.
5. A radio station in Hawaii has hidden a
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.
The second day the clue says that the
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 ∠𝑃 angle with ray 𝑃𝐶⃗⃗⃗⃗ ⃗, show your construction marks.
a) Label point C on the angle
bisector
b) Construct the perpendicular
from point C to each ray of ∠𝑃
P c) Label the intersections B and K
d) Measure 𝐶𝐵 𝑎𝑛𝑑 𝐶𝐾
e) What do you notice?
2) Measure ∠D
a) Measure an equal distance (in cm)
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
D
e) Draw 𝐷𝑇⃗⃗⃗⃗ ⃗
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
1) Find the median from vertex 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
G
P
I
Name: ____________________________Period: ________ 5.4 Medians and Centroids
In the space below do the following constructions:
1) Construct a large triangle ΔRST, use compass & straight edge