Download Optional extra problems

Name
March 24, 2017
Math 2 practice test
coordinate geometry page 1
At our next class, we will have a test on coordinate geometry. It will be a 30-40 minute test that
counts with three-fourths the weight of a full period test. To help you review for the quest,
today’s set of problems is a practice test that we will work on. Use your group partners as
needed, but remember the more you rely on them the less you can do independently.
General instructions: Wherever you are asked to prove or support your answers, use coordinate
methods (such as calculating slopes, distances, etc.).
1. Here are equations of two lines:
y=
3
4
(x + 1) + 5
y = – 23 (x + 1) + 5
a. What point do these two lines have in common? Briefly explain how you know.
b. Write equations for two more lines, such that the four lines form a parallelogram.
c. Determine whether or not your parallelogram is a rhombus.
Give evidence supporting your answer.
Name
March 24, 2017
Math 2 practice test
coordinate geometry page 2
2. a. Prove that the quadrilateral
shown here is an isosceles
trapezoid.
b. Prove that the diagonals of this quadrilateral are congruent.
c. Do the diagonals bisect each other? Prove or disprove.
Name
March 24, 2017
Math 2 practice test
coordinate geometry page 3
3. Suppose that a quadrilateral has one
diagonal that perpendicularly bisects the
other diagonal. Shown is an appropriate
coordinate setup for that situation.
a. Which diagonal is the one that is
bisected?
b. Calculate the side lengths of the
quadrilateral. What special kind of
quadrilateral is this? Prove it.
c. Under what conditions (concerning a, b, and/or c) would this quadrilateral be a square?
For the conditions that you have specified, prove that the quadrilateral is a square.
Name
March 24, 2017
Math 2 practice test
coordinate geometry page 4
Answers
1. a. (–1, 5)
b. Write another line with a slope of
3
4
and another line with a slope of  23 .
c. Answer varies depending on your choice of lines, but you need to have calculated the
lengths of the sides using the distance formula (or the Pythagorean Theorem). If all the
lengths are equal then the answer is yes; otherwise the answer is no.
2. a. The quadrilateral is a trapezoid because AB and DC both have slope –1, and it’s
isosceles because AD and BC both have length 10 .
b. AC and BD both have length
80 .
c. No, because their midpoints aren’t the same: AC has midpoint (1, 0) but BD has
midpoint (2, –1).
3. a. the diagonal from (–a, 0) to (a, 0), because (0, 0) is its midpoint.
b. The upper two side lengths simplify to
simplify to
a 2  b 2 while the lower two side lengths
a 2  c 2 so the quadrilateral is a kite.
c. The conditions are that a = b = –c. If so, then:



All the side lengths are equal because they all simplify to a 2  b 2 .
The slopes of the sides are 1 and –1, which are opposite reciprocals, so the sides are
perpendicular and all of the angles are right angles.
Since the quadrilateral has all sides equal and all right angles, it is a square.
Optional extra problems
4. Consider the line y = –3(x – 2) – 1.
a. If you were working with the slope of this line and you had run=4, what would the rise be?
b. Explain why this line includes the point (2, –1).
c. Write an equation for a line that is perpendicular to the given line and also includes
the point (2, –1).
d. Write an equation for a line that is perpendicular to the given line but does not include
the point (2, –1).
5. For the quadrilateral whose vertices are (–1, –1), (0, 3), (4, 2), and (3, –2), prove each of the
following using coordinate methods.
a. Prove that it is a parallelogram.
b. Prove that it is a rectangle.
c. Prove that it is a rhombus.
d. Prove that it is a square.
e. Prove that the diagonals are congruent.
f. Prove that the diagonals are perpendicular.