Unit 3-Aim 3: How can we prove a quadrilateral is a trapezoid?
Motivation:
1. What do the following pictures have in common?
Answer: They contain trapezoids (fill in the aim).
2. What is a trapezoid?
Answer: A trapezoid is a quadrilateral in which only two sides are parallel.
3a. Given: A(1,1), B(2,5), C(5,7), and D(7,5), how can we prove that ABCD is a trapezoid?
Proving quadrilaterals are trapezoids
1. Find the slopes of all four sides.
2. Only two slopes are equal, so only
two sides are parallel.
3b. Given: A(1,1), B(2,5), C(5,7), and D(7,5)
Prove: ABCD is a trapezoid. [The use of the accompanying grid is optional.]
Read the question carefully. The
word trapezoid, by definition, tells
you that you are looking for a figure
with ONLY ONE set of parallel
sides. Lines are parallel when they
have the same slope. You will use
the Slope Formula.
Draw a neat, labeled graph for the
problem.
State the formula that you will be
using.
Show ALL work!! We are looking
for ONE set of parallel sides AND
one set on non-parallel sides.
Be sure to state the connection
between the slopes and the sides
being parallel or non-parallel.
End with a concluding sentence
stating WHY you know the figure is
a trapezoid.
4. B 8/01: 34 Given: A(1,6), B(7,9), C(13,6), and D(3,1)
Prove: ABCD is a trapezoid. [The use of the accompanying grid is optional.]
5. Quadrilateral OPQR has vertices O(0,0), P(9,3), Q(9,8), and R(–3,4).
a. Prove that OPQR is a trapezoid. [The use of the grid is optional.]
b. What is an isosceles trapezoid ?
Answer: An isosceles trapezoid is a trapezoid in which the non-parallel sides are congruent.
c. How can we prove that OPQR is an isosceles trapezoid?
Answer:
Proving trapezoids are isosceles
1. Find the two distances of the non-parallel sides.
2. If the non-parallel sides are congruent (equal in
distance), then the trapezoid is isosceles.
or
1. Find the two distances of the diagonals.
2. If the diagonals are congruent (equal in distance), then
the trapezoid is isosceles.
d. Prove that OPQR is an isosceles trapezoid.
PI GG69 Lesson #57
AIM: How can we prove other specific quadrilateral relationships using coordinate geometry?
Students will be able to:
1. given the coordinates of the vertices of a quadrilateral, apply coordinate geometry methods to show that
quadrilateral is a rhombus, square, trapezoid or isosceles trapezoid and justify their conclusion
2. prove, given the specific coordinates of the vertices of a quadrilateral that the diagonals bisect each other and/or
are congruent and/or are perpendicular to each other and justify their conclusion
3. given the coordinates of the vertices of a quadrilateral, prove that a median is parallel to either base and/or is
equal in length to one-half the sum of the bases
Writing Exercise:
Describe in your own words, how the distance, slope, and midpoint formulas help to demonstrate the properties of
geometric figures.
Student Name ____________________________
Unit 3-Aim 3: How can we prove a quadrilateral is a ______________________?
1. What do the following pictures have in common?
_____________________________________________________________________________________
_____________________________________________________________________________________
2. What is a ___________________?
_____________________________________________________________________________________
3a. Given: A(1,1), B(2,5), C(5,7), and D(7,5), how can we prove that ABCD is a trapezoid?
Proving quadrilaterals are trapezoids
1. Find the ________ of all four sides.
2. _______ two _______ are _________, so only
______ _______ are _______________.
3b. Given: A(1,1), B(2,5), C(5,7), and D(7,5)
Prove: ABCD is a trapezoid. [The use of the accompanying grid is optional.]
Read the question carefully. The
word trapezoid, by definition, tells
you that you are looking for a figure
with ONLY ONE set of parallel
sides. Lines are parallel when they
have the same slope. You will use
the Slope Formula.
Draw a neat, labeled graph for the
problem.
State the formula that you will be
using.
Show ALL work!! We are looking
for ONE set of parallel sides AND
one set on non-parallel sides.
Be sure to state the connection
between the slopes and the sides
being parallel or non-parallel.
End with a concluding sentence
stating WHY you know the figure is
a trapezoid.
4. Given: A(1,6), B(7,9), C(13,6), and D(3,1)
Prove: ABCD is a trapezoid. [The use of the accompanying grid is optional.]
5. Quadrilateral OPQR has vertices O(0,0), P(9,3), Q(9,8), and R(–3,4).
a. Prove that OPQR is a trapezoid. [The use of the grid is optional.]
b. What is an isosceles trapezoid ?
_____________________________________________________________________________________
_____________________________________________________________________________________
c. How can we prove that OPQR is an isosceles trapezoid?
Proving trapezoids are isosceles
1. Find the two ______________of the non-parallel sides.
2. If the non-parallel sides are congruent (equal in
distance), then the trapezoid is isosceles.
or
1. Find the two _____________ of the diagonals.
2. If the diagonals are congruent (equal in distance), then
the trapezoid is isosceles.
d. Prove that OPQR is an isosceles trapezoid.
Extra:
Quadrilateral KATE has vertices K(1,5), A(4,7), T(7,3), and E(1,–1).
a. Prove that KATE is a trapezoid. [The use of the grid is optional.]
b. Prove that KATE is not an isosceles trapezoid.
HW5 Quadrilateral KATE has vertices K(  3,  1), A(  1,3), T(3,4), and E(5,1).
a. Prove that KATE is a trapezoid. [The use of the grid is optional.]
b. Prove that KATE is not an isosceles trapezoid.
HW 4 Course II Jan 03: 42 The vertices of quadrilateral BIRD are B(–1,–3), I(8,0), R(3,5), and D(0,4).
Prove by means of coordinate geometry that quadrilateral BIRD is an isosceles trapezoid. [10]
HW4 Course II June 01: 39 The vertices of  PQR are P(8,6), Q(–1,13), and R(5,–5).
a Find the coordinates of M, the midpoint of QR . [2]
b Prove that PM is perpendicular to QR . [4]
c Prove that  PQR is isosceles. [4]
38 Triangle ABC has vertices A(–2,1), B(5,5), and C(–1,–7).
a Show that  ABC is isosceles. [4]
b Find the length of the altitude drawn from A to BC . [4]
Test3 Course II June 00: 37 The coordinates of the vertices of  ABC are A(–4,1), B(4,9), and C(9,–
2). Point M(1,6) lies on AB .
a Show by means of coordinate geometry that CM  AB . [4]
A rancher is putting up a new fenced in area for his horses. The corners of the fenced area are located at
A(0,0), B(20,0), C(8,24) and D(-8,16).
a. By checking slopes, show the relationship between AD and BC .
b. By checking slopes, show the relationship between AD , BC and DC .
c. What kind of quadrilateral is the pasture, and why?
Test3 Quadrilateral LISA has vertices L(0, 0), I(5, 0), S(2, 6), and A(  2, 4).
a. Prove that LISA is a trapezoid. [The use of the grid is optional.]
b. Prove that LISA is not an isosceles trapezoid.