MATH 3395 Construction Project
The following construction problems should be worked on and solved in groups of three
people. Each group will be assessed three times—once for each category. When your
group is ready to be assessed, notify me and indicate on which category the group wants to
be assessed. Your group will choose the problem in that category on which to be assessed.
The group member who will be asked to solve the problem will be selected randomly.
Whatever grade is assigned to that student will be the grade for the group. Be sure each
group member can solve each problem correctly. The following scoring guidelines will be
used to assess each problem. The maximum possible point total is 45 points.
0
3
6
Construction
Cannot do the construction correctly even with hints
Can do the construction correctly but only after hints are given
Can do the construction correctly without hints
0
3
6
Proof
Cannot do the proof correctly even with hints
Can do the proof correctly but only after hints are given
Can do the proof correctly without hints
1. Construct a rhombus (not a square), given
a. one side and one angle (1 point)
b. one angle and a diagonal (2 points)
c. the altitude and one diagonal (3 points)
2. Construct a parallelogram (not a rhombus), given
a. one side, one angle, and one diagonal (1 point)
b. two adjacent sides and an altitude (2 points)
c. one angle, one side, and the altitude on that side (3 points)
3. Construct an isosceles trapezoid, given
a. the diagonal, altitude, and one of the bases (1 point)
b. one base, the diagonal, and the angle included by them (2 points)
c. the bases and one angle (3 points)
Figures to use as the given information
angle
side or base
side or altitude
base or diagonal
SAMPLE
Construct a rhombus given one side and an altitude
Solution
1. Begin by copying one of the sides on the “Figures” sheet and label it AB .
2. Choose a point P between A and B and construct a perpendicular to AB through P.
3. Copy the altitude from the “figures” sheet so that one its endpoints is point P and the
other is on the perpendicular you constructed. Label the other endpoint Q.
4. Using point A as center, make a circle with radius AB.
5. Construct a line through point Q perpendicular to PQ (construction marks not shown).
Label the point where it intersects the circle point R.
6. Construct radius AR .
7. Using point R as center, mark off a length equal to AB along ray QR. Call the intersection point T.
8. Construct BT .
Q
A
R
T
B
P
Conclusion: ARTB is a rhombus with
the required information.
Brief Proof:
Because AR, AB, and RT, are all radii of
equal circles, they have the same length.
Because RT and AB are both perpendicular
to PQ, RT is parallel to AB. Therefore,
ARTB a parallelogram (one pair of opposite
sides are both congruent and parallel). It is
now also a rhombus because two adjacent
sides are congruent.
The sides of the rhombus are all congruent to
AB, which was the given side. Since QT is
parallel to AB, the distance from QT to AB
is always the same, and that distance is equal
to PQ, the given altitude.