Proofs with Coordinates
Chapter 11
Lesson
11-4
Vocabulary
Proofs with Coordinates
convenient location for a figure
BIG IDEA Coordinate proofs are like other proofs, but start with a figure graphed on a coordinate plane.
So far, almost every proof in this book has been in synthetic geometry, where points are locations. Because figures in coordinate geometry have the same properties as those in synthetic geometry, it is not surprising that there are proofs in coordinate geometry. They have conclusions and justifications just like other proofs. However, because the proofs use coordinates, the justifications are often from algebra or arithmetic.
Mental Math A sphere’s radius is greater than 4 inches. What can you conclude about a. the area of a great circle of the sphere? b. the sphere’s surface area?
Recall that the slope m of a line determined by two points (x1, y1) and y 2 - y1 (x 2, y 2) is defined as m = ______ x 2 - x 1 . Recall also the Parallel Lines and Slopes Theorem: Two nonvertical lines are parallel if and only if they have the same slope. These ideas are utilized in Example 1.
c. the sphere’s volume ?
Example 1 Consider quadrilateral ABCD with vertices A = (3, 3), B = (5, 10), C = (15, 10), and D = (13, 3). Prove that ABCD is a parallelogram. Solution Draw and label a picture, as done at the right.
10
y
B =(5, 10)
C =(15, 10)
In the drawing, it appears that ABCD is a parallelogram. ABCD is a parallelogram if both pairs of opposite sides are parallel. Recall that two lines are parallel if they have the same slope. So calculate the slopes of the sides of ABCD. To stress the justification of each step, this proof is written in two-column form.
670
5
A =(3, 3)
D =(13, 3) x
0
5
10
15
Indirect Proofs and Coordinate Proofs
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Lesson 11-4 Conclusions −− 3-3 0 1. slope of AD = _____ = __ = 0 13 - 3 10 −− ______ 10 - 10 0 slope of BC = = __ 15 - 5 10 −− 7 10 - 3 slope of DC = ______ = _ 2 15 - 13 −− ______ 7 10 - 3 __ slope of AB = 5 - 3 = 2
−− 2. AD
−− −− −− BC , DC AB.
3. ABCD is a parallelogram.
Justifications 1. definition of slope
=0
2. Parallel Lines and Slopes Theorem 3. definition of parallelogram
The proof in Example 1 is about a single parallelogram. To prove a theorem that applies to all parallelograms, the coordinates of the vertices must be variables.
Example 2 Prove that the diagonals of a square SQRE with S = (– a, – a), Q = (– a, a), R = (a, a), and E = (a, – a) are perpendicular.
y
Q =(᎑a, a)
R =(a, a)
Proof The Perpendicular Lines and Slopes Theorem tells us that the diagonals are perpendicular if the product of their slopes is –1. Calculate the slopes of the diagonals: −− a - (– a) a+a 2a ___ Slope of SR = _______ = _____ a + a = 2a = 1 a - (– a) −− a - (– a) a+a 2a _____ ___ Slope of EQ = _______ – a - a = –2a = –2a = –1
x
S =(᎑a, ᎑a)
E =(a, ᎑a)
The product of the slopes is 1 · –1, or –1. The diagonals of SQRE are perpendicular by the Perpendicular Lines and Slopes Theorem.
The last statement of the proof in Example 2 is imperative. Without it, your proof is like an incomplete sentence. You must justify why your calculations prove the statement that is to be proved.
Convenient Locations The coordinates of the vertices of the square in Example 2 were carefully chosen to be representative of a general square. By using variables as coordinates, the length of the side of the square is arbitrary. But the coordinates of the square were chosen such that the center is the origin. Still, the above proof holds for any square; not just those centered at the origin. Why? Because no matter where a square is located, a coordinate plane can be established with the origin at the center of the square and the x-axis and y-axis each parallel to two sides of the square. Proofs with Coordinates
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Chapter 11
The coordinates given in Example 2 are said to be a convenient location for the square. A convenient location for a figure is one in which its key points are described with the least number of different variables. In Example 2, the vertices of the square are described with just one variable, a. This was possible because all sides of a square share one relationship: The lengths are the same. Additional variables are necessary when there are multiple relationships among sides of the polygon. For example, in a general rectangle, two pairs of opposite sides are congruent, but they are not all congruent to each other. Thus, two variables are necessary.
QY1
In the rectangle below, each horizontal side has length a and each vertical side has length b. Complete the convenient locations of its vertices. y ? (0, ___)
(___ ? , ___) ?
QY1 x
The location in Example 2 turns out to be convenient because the square is rotation-symmetric with center of rotation (0, 0). When a polygon is reflection-symmetric, a convenient location can usually be found by locating the polygon so that it is symmetric to the x-axis or the y-axis. Otherwise, a convenient location can be found by placing the polygon with one vertex at (0, 0) and another vertex on one of the axes.
To remember convenient locations, recall how to find certain reflection and rotation images on a coordinate plane. The reflection image of (a, b) over the x-axis is (a, –b). The reflection image of (a, b) over the y-axis is (– a, b). The image of (a, b) under a rotation of 180º about the origin is (– a, – b). The point (a, b) and the three images (a, –b), (– a, b) and (– a, – b) are the vertices of a rectangle. This is another convenient location for a rectangle. Here are some convenient locations for some of the figures you have studied. parallelogram (two convenient locations)
(b, c)
(a + b, c)
(0, 0)
(a, 0)
(᎑c, ᎑b)
x (᎑a, 0)
(a, 0)
(0, ᎑c) right triangle y
isosceles triangle y
(0, b)
(0, b)
(0, 0)
rectangle (two convenient locations)
(c, b)
(᎑a, ᎑b)
x
x (0, 0) (a, 0)
(᎑a, 0)
y
(᎑a, b) x
x
672
Name three other types of quadrilaterals that are rotation-symmetric.
(0, b)
y
(a, b)
? , 0) (___
QY2
kite y
QY2
y
(0, 0)
(0, b)
(a, b)
(0, 0)
(a, 0)
(a, 0)
y (a, b) x
x
(᎑a, ᎑b)
(a, ᎑b)
Indirect Proofs and Coordinate Proofs
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Lesson 11-4
Notice how the convenient locations take advantage of the symmetry of the polygons. If a polygon is rotation-symmetric, like the parallelogram, then the origin can be at the center of the rotation. If a polygon is reflection-symmetric, like the kite, then the line of symmetry can be placed along either the x-axis or the y-axis. QY3
QY3
Once you have a figure in a convenient location, coordinate proofs of parallelism and perpendicularity can be rather straightforward. Just calculate and compare slopes and remember to justify how the slopes help to prove or disprove your statement.
Questions
Use the convenient location for the parallelogram, with one vertex at (0, 0), to show that the opposite sides of a parallelogram are parallel.
COVERING THE IDEAS 1. Fill in the Blank To prove a theorem that applies to all figures of a given type, the nonzero coordinates of the vertices must ? be . 2. Create a convenient location for a square that has one vertex at the origin, one side on the x-axis, and a side of length s. 3. In the lesson, a convenient location is shown for a kite whose symmetry diagonal lies on the y-axis. Create a convenient location for a kite whose symmetry diagonal is on the x-axis. 4. If C = (9, 7), N = (3, 6), J = (–3, 2) and O = (4, 3), prove that CNJO is not a trapezoid. y 10
C =(9, 7)
N =(3, 6) 5
J =(᎑3, 2)
O =(4, 3) x
᎑5
0
5
10
In 5 and 6, draw a figure of the indicated type in a convenient location on a coordinate plane.
5. right triangle with legs of length 14 and 17 6. isosceles triangle that is symmetric to the x-axis
Proofs with Coordinates
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Chapter 11
7. MARK is a rhombus with vertices M = (– 3, 0), A = (–2, 5), R = (3, 6) and K = (2, 1). Prove that the diagonals of MARK are perpendicular. 8. A rhombus in a convenient position has vertices (a, 0), (0, b), (a, 0), and (0, –b). a. Draw such a rhombus. b. Explain why the diagonals of the rhombus are perpendicular.
10
A =(᎑2, 5)
y
R =(3, 6) 5
K =(2, 1)
M =(᎑3, 0) ᎑5
x
0
5
APPLYING THE MATHEMATICS 9. Give the vertices of a general isosceles trapezoid in a convenient location that is symmetric to the y-axis. 10. Given: Quadrilateral KAYL, with K = (– c, –d), A = (–a, b), Y = (c, d), and L = (a, –b), where a + c ≠ 0. Prove: KAYL is a parallelogram. y
11. Fill in the missing coordinate for the convenient location for the equilateral triangle at the right.
? (0, ___)
12. Prove that the diagonals of a kite are perpendicular. In 13 and 14, consider quadrilateral SANG with S = (0.3, – 0.3), A = (0.4, – 0.1), N = (– 0.2, 0.2), and G = (– 0.3, 0).
13. a. Find the image of SANG under a size change of magnitude 10 and center at the origin. Call the image SANG. b. Prove that SANG is a parallelogram. c. From this, explain why SANG must be a parallelogram. 14. Prove: SANG is a rectangle.
x
(᎑a, 0)
0
(a, 0)
15. Tiana and Angel are on a treasure hunt. Angel’s map says, “Go 3 steps north and 2 steps east.” Tiana starts 4 steps south of Angel. Her map says, “Go 3 steps south and 2 steps west.” Describe the figure formed by connecting the ending point of each girl to each starting point. 16. Consider the figure below. Find an equation for the line containing point B that would create a rectangle BCDE. D =(᎑3, 5)
y 5
E = (᎑ 4, 3) B = (0, 1) ᎑5
674
0
x 5
Indirect Proofs and Coordinate Proofs
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Lesson 11-4
REVIEW 17. The famous baseball player Yogi Berra was also famous for many memorable quotations. He once said: “If you can’t imitate him, don’t copy him.” Write down the converse, the inverse, and the contrapositive of this statement. (Lesson 11-2) 18. In Lewis Carroll’s book Alice’s Adventures in Wonderland, the following line appears: “Contrariwise,” continued Tweedledee, “if it was so, it might be; and if it were so, it would be; but as it isn’t, it ain’t. That’s logic.” What principle of logic is Tweedledee using in his argument? (Lesson 11-1) 19. One box contains a sphere of radius 2, and another box contains two spheres each of radius 1. If both boxes are as small as possible, which has the greater volume? (Lesson 10-7)
Yogi Berra played for the New York Yankees from 1946 to 1963, usually as a catcher.
20. Suppose Ty, Rachelle, and Makayla all live on the same side of the street. Ty lives between Rachelle and Makayla. He lives 350 yards from Makayla, and Makayla and Rachelle live 975 yards apart. (Lesson 1-6) a. Picture the situation on a number line and label the given distances. b. How far does Ty live from Rachelle? 21. Full-grown zebras can range from 46 to 55 inches high at the shoulder and their weights can range from 550 to 650 pounds. Let h be these possible heights and w be these possible weights. (Previous Course)
a. Graph all possible ordered pairs (h, w). b. Describe the graph. √ x4 22. Simplify the expression ____. (Previous Course) √ x2 23. Simplify the expression (–(x − y))2 − (x + y)2. (Previous Course)
EXPLORATION 24. Three vertices of a parallelogram are (–2, 0), (2, 3), and (3, –1). a. Find at least two possible locations of the fourth vertex. b. Are there other possible locations? Explain. = – __1 x + 2, CD = 2x, 25. The equations of three lines are given BC 2
QY ANSWERS
1. first quadrant rectangle with vertices (0, 0), (a, 0), (a, b), and (0, b) 2. parallelogram, rectangle, rhombus c-0
c
_ 3. The slopes are _____ b - 0 = b, c-0 c _____ 0-0 _________ _ (a + b) - a = b , a - 0 = 0, c-c and ________ (a + b) – b = 0. Opposite
= – __1 x − 2. Find an equation for AB so that ABCD is and DA
sides have the same slope, so
a rectangle.
they are parallel.
2
Proofs with Coordinates
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Lesson
11-4
Vocabulary
Proofs with Coordinates
convenient location for a figure
BIG IDEA Coordinate proofs are like other proofs, but start with a figure graphed on a coordinate plane.
So far, almost every proof in this book has been in synthetic geometry, where points are locations. Because figures in coordinate geometry have the same properties as those in synthetic geometry, it is not surprising that there are proofs in coordinate geometry. They have conclusions and justifications just like other proofs. However, because the proofs use coordinates, the justifications are often from algebra or arithmetic.
Mental Math A sphere’s radius is greater than 4 inches. What can you conclude about a. the area of a great circle of the sphere? b. the sphere’s surface area?
Recall that the slope m of a line determined by two points (x1, y1) and y 2 - y1 (x 2, y 2) is defined as m = ______ x 2 - x 1 . Recall also the Parallel Lines and Slopes Theorem: Two nonvertical lines are parallel if and only if they have the same slope. These ideas are utilized in Example 1.
c. the sphere’s volume ?
Example 1 Consider quadrilateral ABCD with vertices A = (3, 3), B = (5, 10), C = (15, 10), and D = (13, 3). Prove that ABCD is a parallelogram. Solution Draw and label a picture, as done at the right.
10
y
B =(5, 10)
C =(15, 10)
In the drawing, it appears that ABCD is a parallelogram. ABCD is a parallelogram if both pairs of opposite sides are parallel. Recall that two lines are parallel if they have the same slope. So calculate the slopes of the sides of ABCD. To stress the justification of each step, this proof is written in two-column form.
670
5
A =(3, 3)
D =(13, 3) x
0
5
10
15
Indirect Proofs and Coordinate Proofs
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Lesson 11-4 Conclusions −− 3-3 0 1. slope of AD = _____ = __ = 0 13 - 3 10 −− ______ 10 - 10 0 slope of BC = = __ 15 - 5 10 −− 7 10 - 3 slope of DC = ______ = _ 2 15 - 13 −− ______ 7 10 - 3 __ slope of AB = 5 - 3 = 2
−− 2. AD
−− −− −− BC , DC AB.
3. ABCD is a parallelogram.
Justifications 1. definition of slope
=0
2. Parallel Lines and Slopes Theorem 3. definition of parallelogram
The proof in Example 1 is about a single parallelogram. To prove a theorem that applies to all parallelograms, the coordinates of the vertices must be variables.
Example 2 Prove that the diagonals of a square SQRE with S = (– a, – a), Q = (– a, a), R = (a, a), and E = (a, – a) are perpendicular.
y
Q =(᎑a, a)
R =(a, a)
Proof The Perpendicular Lines and Slopes Theorem tells us that the diagonals are perpendicular if the product of their slopes is –1. Calculate the slopes of the diagonals: −− a - (– a) a+a 2a ___ Slope of SR = _______ = _____ a + a = 2a = 1 a - (– a) −− a - (– a) a+a 2a _____ ___ Slope of EQ = _______ – a - a = –2a = –2a = –1
x
S =(᎑a, ᎑a)
E =(a, ᎑a)
The product of the slopes is 1 · –1, or –1. The diagonals of SQRE are perpendicular by the Perpendicular Lines and Slopes Theorem.
The last statement of the proof in Example 2 is imperative. Without it, your proof is like an incomplete sentence. You must justify why your calculations prove the statement that is to be proved.
Convenient Locations The coordinates of the vertices of the square in Example 2 were carefully chosen to be representative of a general square. By using variables as coordinates, the length of the side of the square is arbitrary. But the coordinates of the square were chosen such that the center is the origin. Still, the above proof holds for any square; not just those centered at the origin. Why? Because no matter where a square is located, a coordinate plane can be established with the origin at the center of the square and the x-axis and y-axis each parallel to two sides of the square. Proofs with Coordinates
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Chapter 11
The coordinates given in Example 2 are said to be a convenient location for the square. A convenient location for a figure is one in which its key points are described with the least number of different variables. In Example 2, the vertices of the square are described with just one variable, a. This was possible because all sides of a square share one relationship: The lengths are the same. Additional variables are necessary when there are multiple relationships among sides of the polygon. For example, in a general rectangle, two pairs of opposite sides are congruent, but they are not all congruent to each other. Thus, two variables are necessary.
QY1
In the rectangle below, each horizontal side has length a and each vertical side has length b. Complete the convenient locations of its vertices. y ? (0, ___)
(___ ? , ___) ?
QY1 x
The location in Example 2 turns out to be convenient because the square is rotation-symmetric with center of rotation (0, 0). When a polygon is reflection-symmetric, a convenient location can usually be found by locating the polygon so that it is symmetric to the x-axis or the y-axis. Otherwise, a convenient location can be found by placing the polygon with one vertex at (0, 0) and another vertex on one of the axes.
To remember convenient locations, recall how to find certain reflection and rotation images on a coordinate plane. The reflection image of (a, b) over the x-axis is (a, –b). The reflection image of (a, b) over the y-axis is (– a, b). The image of (a, b) under a rotation of 180º about the origin is (– a, – b). The point (a, b) and the three images (a, –b), (– a, b) and (– a, – b) are the vertices of a rectangle. This is another convenient location for a rectangle. Here are some convenient locations for some of the figures you have studied. parallelogram (two convenient locations)
(b, c)
(a + b, c)
(0, 0)
(a, 0)
(᎑c, ᎑b)
x (᎑a, 0)
(a, 0)
(0, ᎑c) right triangle y
isosceles triangle y
(0, b)
(0, b)
(0, 0)
rectangle (two convenient locations)
(c, b)
(᎑a, ᎑b)
x
x (0, 0) (a, 0)
(᎑a, 0)
y
(᎑a, b) x
x
672
Name three other types of quadrilaterals that are rotation-symmetric.
(0, b)
y
(a, b)
? , 0) (___
QY2
kite y
QY2
y
(0, 0)
(0, b)
(a, b)
(0, 0)
(a, 0)
(a, 0)
y (a, b) x
x
(᎑a, ᎑b)
(a, ᎑b)
Indirect Proofs and Coordinate Proofs
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Lesson 11-4
Notice how the convenient locations take advantage of the symmetry of the polygons. If a polygon is rotation-symmetric, like the parallelogram, then the origin can be at the center of the rotation. If a polygon is reflection-symmetric, like the kite, then the line of symmetry can be placed along either the x-axis or the y-axis. QY3
QY3
Once you have a figure in a convenient location, coordinate proofs of parallelism and perpendicularity can be rather straightforward. Just calculate and compare slopes and remember to justify how the slopes help to prove or disprove your statement.
Questions
Use the convenient location for the parallelogram, with one vertex at (0, 0), to show that the opposite sides of a parallelogram are parallel.
COVERING THE IDEAS 1. Fill in the Blank To prove a theorem that applies to all figures of a given type, the nonzero coordinates of the vertices must ? be . 2. Create a convenient location for a square that has one vertex at the origin, one side on the x-axis, and a side of length s. 3. In the lesson, a convenient location is shown for a kite whose symmetry diagonal lies on the y-axis. Create a convenient location for a kite whose symmetry diagonal is on the x-axis. 4. If C = (9, 7), N = (3, 6), J = (–3, 2) and O = (4, 3), prove that CNJO is not a trapezoid. y 10
C =(9, 7)
N =(3, 6) 5
J =(᎑3, 2)
O =(4, 3) x
᎑5
0
5
10
In 5 and 6, draw a figure of the indicated type in a convenient location on a coordinate plane.
5. right triangle with legs of length 14 and 17 6. isosceles triangle that is symmetric to the x-axis
Proofs with Coordinates
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Chapter 11
7. MARK is a rhombus with vertices M = (– 3, 0), A = (–2, 5), R = (3, 6) and K = (2, 1). Prove that the diagonals of MARK are perpendicular. 8. A rhombus in a convenient position has vertices (a, 0), (0, b), (a, 0), and (0, –b). a. Draw such a rhombus. b. Explain why the diagonals of the rhombus are perpendicular.
10
A =(᎑2, 5)
y
R =(3, 6) 5
K =(2, 1)
M =(᎑3, 0) ᎑5
x
0
5
APPLYING THE MATHEMATICS 9. Give the vertices of a general isosceles trapezoid in a convenient location that is symmetric to the y-axis. 10. Given: Quadrilateral KAYL, with K = (– c, –d), A = (–a, b), Y = (c, d), and L = (a, –b), where a + c ≠ 0. Prove: KAYL is a parallelogram. y
11. Fill in the missing coordinate for the convenient location for the equilateral triangle at the right.
? (0, ___)
12. Prove that the diagonals of a kite are perpendicular. In 13 and 14, consider quadrilateral SANG with S = (0.3, – 0.3), A = (0.4, – 0.1), N = (– 0.2, 0.2), and G = (– 0.3, 0).
13. a. Find the image of SANG under a size change of magnitude 10 and center at the origin. Call the image SANG. b. Prove that SANG is a parallelogram. c. From this, explain why SANG must be a parallelogram. 14. Prove: SANG is a rectangle.
x
(᎑a, 0)
0
(a, 0)
15. Tiana and Angel are on a treasure hunt. Angel’s map says, “Go 3 steps north and 2 steps east.” Tiana starts 4 steps south of Angel. Her map says, “Go 3 steps south and 2 steps west.” Describe the figure formed by connecting the ending point of each girl to each starting point. 16. Consider the figure below. Find an equation for the line containing point B that would create a rectangle BCDE. D =(᎑3, 5)
y 5
E = (᎑ 4, 3) B = (0, 1) ᎑5
674
0
x 5
Indirect Proofs and Coordinate Proofs
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Lesson 11-4
REVIEW 17. The famous baseball player Yogi Berra was also famous for many memorable quotations. He once said: “If you can’t imitate him, don’t copy him.” Write down the converse, the inverse, and the contrapositive of this statement. (Lesson 11-2) 18. In Lewis Carroll’s book Alice’s Adventures in Wonderland, the following line appears: “Contrariwise,” continued Tweedledee, “if it was so, it might be; and if it were so, it would be; but as it isn’t, it ain’t. That’s logic.” What principle of logic is Tweedledee using in his argument? (Lesson 11-1) 19. One box contains a sphere of radius 2, and another box contains two spheres each of radius 1. If both boxes are as small as possible, which has the greater volume? (Lesson 10-7)
Yogi Berra played for the New York Yankees from 1946 to 1963, usually as a catcher.
20. Suppose Ty, Rachelle, and Makayla all live on the same side of the street. Ty lives between Rachelle and Makayla. He lives 350 yards from Makayla, and Makayla and Rachelle live 975 yards apart. (Lesson 1-6) a. Picture the situation on a number line and label the given distances. b. How far does Ty live from Rachelle? 21. Full-grown zebras can range from 46 to 55 inches high at the shoulder and their weights can range from 550 to 650 pounds. Let h be these possible heights and w be these possible weights. (Previous Course)
a. Graph all possible ordered pairs (h, w). b. Describe the graph. √ x4 22. Simplify the expression ____. (Previous Course) √ x2 23. Simplify the expression (–(x − y))2 − (x + y)2. (Previous Course)
EXPLORATION 24. Three vertices of a parallelogram are (–2, 0), (2, 3), and (3, –1). a. Find at least two possible locations of the fourth vertex. b. Are there other possible locations? Explain. = – __1 x + 2, CD = 2x, 25. The equations of three lines are given BC 2
QY ANSWERS
1. first quadrant rectangle with vertices (0, 0), (a, 0), (a, b), and (0, b) 2. parallelogram, rectangle, rhombus c-0
c
_ 3. The slopes are _____ b - 0 = b, c-0 c _____ 0-0 _________ _ (a + b) - a = b , a - 0 = 0, c-c and ________ (a + b) – b = 0. Opposite
= – __1 x − 2. Find an equation for AB so that ABCD is and DA
sides have the same slope, so
a rectangle.
they are parallel.
2
Proofs with Coordinates
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