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Name: _________________________________________________
Period: ____________
Homework Set #2– 5th Term (Due at the beginning of class Friday, March 28th)
This assignment is worth 16 points (due to the 4-day school week)
Tuesday night: p.231 #6-9 part b only
-complete at least one problem from HW set
Wednesday night: p.231 #10-11
-complete a second problem from HW set
Thursday night: p.231 #12
-complete homework set
Friday night: p.237 #31 & 32, p.238 #25
Weekly assignment (due at the beginning of class Friday, March 28th):
 Responses must be written on separate paper.
 You may work with others, but your write ups must be in your own words, or a
grade of 0 will be given on the entire assignment for cheating.
1.) (5 points)
Given: XYZ has vertices X: (0,0), Y: (a, 0), and Z: (0,b)
Prove: The circumcenter is located at the midpoint of the hypotenuse (the midpoint of YZ )
[Hint: You will need to show the equations of the perpendicular bisectors of at least two
sides.]
2.) (able to complete after Wednesday’s class) (3 points)
Concurrent lines
Perpendicular bisectors
Medians (Cevians)
Altitudes
Point of concurrency
Properties
Distance to a vertex is twice the
distance from the circumcenter
to the opposite edge
Angle bisectors
3.) (4 points) Given: SQ and RP are medians of RST ,
RQ=7x-1, SP=5x-4, and QT=6x+9
Prove: PT=46
4.) (3 points) DE is a midsegment of ABC . BF is a median of
ABC . How do DE and AF relate? Justify your reasoning
with appropriate definitions and theorems. [Note: The figure
is not to scale.]
Name: _________________________________________________
Period: ____________
5.) (6 points) Plot ABC with coordinates A: (2,0), B (7,0), and C (5,6). We will verify using
analytic geometry that the circumcenter of this triangle is indeed equidistant from the
vertices.
a.) Plot ABC on graph paper.
b.) Draw in two perpendicular bisectors of the sides. (Since we know that the perpendicular
bisectors are concurrent, we only need two to find the circumcenter.)
c.) What are the equations of the perpendicular bisectors? (You may find using point-slope
form to be easier than slope-intercept form.)
d.) Solve the system of equations you wrote in part c. (You may find using substitution to be
the easiest way.) Remember that your answer should be expressed as a point. This point is
the circumcenter of the triangle—the point of intersection of the perpendicular bisectors.
Label this point D.
e.) Using the point you found in part d, find AD, BD, and CD. Does AD=BD=CD?
6.) (4 points)
Given: O is the circumcenter of ABC .
Prove: AOE  COE
[Note: This could be extended to show that EACH PAIR of inner
triangles are congruent. But, since we showed it for one general
case, it means that it is true for ALL midsegments. ]
7.)
a.)
b.)
c.)
d.)
e.)
(able to do after Tuesday’s class) (6 points)
Plot JKL with J: (-2,1), K: (-2,4), L: (2,0)
Find the equations of two of the perpendicular bisectors.
Find the coordinates of the circumcenter.
Find the equations of two of the altitudes.
Find the coordinates of the orthocenter.
8.)
a.)
b.)
c.)
d.)
(4 points) You will need a compass and straight edge for this question.
Draw several acute triangles. Construct their centroids.
Draw several obtuse triangles. Construct their centroids.
Draw several right triangles. Construct their centroids.
What conjecture can you make about the location of the centroid of a triangle, based on the
size of its angles?
9.) (4 points) (able to do after Tuesday’s class) You will need a compass and straight edge for
this question.
a.) Draw several acute triangles. Construct their orthocenters.
b.) Draw several obtuse triangles. Construct their orthocenters.
c.) Draw several right triangles. Construct their orthocenters.
d.) What conjecture can you make about the location of the orthocenter of a triangle, based on
the size of its angles?