Unit 3-Aim 4: How can we prove coordinate geometry proofs in terms of variables?
Motivation:
1. How is this problem different from others that we have solved?
June 1993: 41 Quadrilateral ABCD has coordinates A0, 0 , B6a, 3b , C3a, 4b , and Da, 3b , a  0
and b  0 .
a. Using coordinate geometry, show that
(1) AB CD (4 points)
(2) AD is not parallel to BC (4 points)
b. Which kind of quadrilateral is ABCD? Why? (2 points)
Answer: The coordinate geometry proof is in terms of variables. (Fill in the aim).
2. Fill in: To graph quadrilateral ABCD
Substitute constants for the variables
3. Solve the problem above.
4. Course II Jan 1987: 42 Quadrilateral QRST has vertices Qa, b , R0, 0 , S c, 0 , and T a  c, b .
Prove that QRST is a parallelogram.
PI GG69 Lesson #58 & 59 (May Require two days)
AIM: How do we use coordinate geometry, where the coordinates are in literal form, to prove relationships for given
geometric figures?
Students will be able to:
1. explore and conjecture how to graph coordinates expressed in literal terms
2. show, by using the distance formula, midpoint formulas, slope, properties of parallel and perpendicular lines, and
equations of lines, that a polygon, given the coordinates of its vertices in literal terms, is a specific type of triangle or
quadrilateral
3. prove that the diagonals of a quadrilateral, given the coordinates of its vertices in literal terms, bisect each other
and/or are congruent and/or are perpendicular to each other
4. give the mathematical justification for the coordinate proof
5. explain, orally and in writing, how to prove a quadrilateral is a specific type of quadrilateral and a triangle is a
specific type of triangle
Writing Exercise:
1. Explain the similarities and differences in proving relationships with numerical examples in comparison to those
with literal examples.
2. How do you know whether or not line segment AB, where A(a ,c) and B(a, d) is a vertical or a horizontal line
segment?
Student Name ____________________________
Unit 3-Aim 4: How can we prove coordinate geometry proofs in terms of ________________________?
1. How is this problem different from others that we have solved?
_____________________________________________________________________________________
_____________________________________________________________________________________
Quadrilateral ABCD has coordinates A0, 0 , B6a, 3b , C3a, 4b , and Da, 3b , a  0 and b  0 .
a. Using coordinate geometry, show that
(1) AB CD (4 points)
(2) AD is not parallel to BC (4 points)
b. Which kind of quadrilateral is ABCD? Why? (2 points)
2. Fill in: To graph quadrilateral ABCD
__________________________ for the variables
3. Solve the problem above.
4. Quadrilateral QRST has vertices Qa, b , R0, 0 , S c, 0 , and T a  c, b . Prove that QRST is a
parallelogram.
Extra:
HW6 Course II June 1987: 42 Quadrilateral JAKE has coordinates J 0, 3a  , A3a, 3a  , K 4a, 0 , and
E a, 0 . Prove by coordinate geometry that quadrilateral JAKE is an isosceles trapezoid.
In the accompanying diagram of ABCD, where a  b, prove ABCD is an isosceles trapezoid.
HW4 B 6/08: 24 The coordinates of quadrilateral PRAT are P(a , b), R ( a , b  3), A(a  3, b  4), and
T (a  6, b  2). Prove that RA is parallel to PT.
HW4 Given: A(0,0), B(a,0), C(a + b, c), and D(b, c)
Prove: ABCD is a parallelogram.
HW5 Given: A(  c ,0), B(a,0), C(a + c, b), and D(0, b)
Prove: ABCD is a parallelogram.
Review Given: A(  b ,  c ), B(a,0), C(b, c), and D(  a , 0)
Prove: ABCD is a parallelogram
Test3 Course II Aug 00: 42 Quadrilateral QUAD has coordinates Q(–a,0), U(3a,0), A(2a,2a), and
D(0,2a). Using coordinate geometry, prove that quadrilateral QUAD is an isosceles trapezoid. [10]
Quadrilateral QUIZ has vertices Q  b, a  , U  0,0  , I  0, c  , and Z  b, a  c  . Prove that QUIZ is a
parallelogram.