In Exercises 13, find the distance between each pair of points. ( 10, 20 ), ( 13, 16 )
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Table of Contents
0
Geometric Art
0.1
Geometry in Nature and in Art
0.2
Line Designs
0.3
Circle Designs
0.4
Op Art
0.5
Knot Designs
0.6
Islamic Tile Designs
1
Introducing Geometry
1.1
Building Blocks of Geometry
1.2
Poolroom Math
1.3
Whats a Widget?
1.4
Polygons
1.5
Triangles
1.6
Special Quadrilaterals
1.7
Circles
1.8
Space Geometry
1.9
A Picture Is Worth a Thousand Words
2
Reasoning in Geometry
2.1
Inductive Reasoning
2.2
Finding the nth Term
2.3
Mathematical Modeling
2.4
Deductive Reasoning
2.5
Angle Relationships
2.6
Special Angles on Parallel Lines
3
3 Using Tools of Geometry
3.1
Duplicating Segments and Angles
3.2
Constructing Perpendicular Bisectors
3.3
Constructing Perpendiculars to a Line
3.4
Constructing Angle Bisectors
3.5
Constructing Parallel Lines
3.6
Construction Problems
3.7
Constructing Points of Concurrency
3.8
The Centroid
4
Discovering and Proving Triangle Properties
4.1
Triangle Sum Conjecture
4.2
Properties of Isosceles Triangles
4.3
Triangle Inequalities
4.4
Are There Congruence Shortcuts?
4.5
Are There Other Congruence Shortcuts?
4.6
Corresponding Parts of Congruent Triangles
4.7
Flowchart Thinking
4.8
Proving Special Triangle Conjectures
5
Discovering and Proving Polygon Properties
5.1
Polygon Sum Conjecture
5.2
Exterior Angles of a Polygon
5.3
Kite and Trapezoid Properties
5.4
Properties of Midsegments
5.5
Properties of Parallelograms
5.6
Properties of Special Parallelograms
5.7
Proving Quadrilateral Properties
6
Discovering and Proving Circle Properties
6.1
Tangent Properties
6.2
Chord Properties
6.3
Arcs and Angles
6.4
Proving Circle Conjectures
6.5
The Circumference/ Diameter Ratio
6.6
Around the World
6.7
Arc Length
7.1
Transformations and Symmetry
7.2
Properties of Isometries
7.3
Compositions of Transformations
7.4
Tessellations with Regular Polygons
7.5
Tessellations with Nonregular Polygons
7.6
Tessellations Using Only Translations
7.7
Tessellations That Use Rotations
7.8
Tessellations That Use Glide Reflections
8
Area
8.1
Areas of Rectangles and Parallelograms
8.2
Areas of Triangles, Trapezoids, and Kites
8.3
Area Problems
8.4
Areas of Regular Polygons
8.5
Areas of Circles
8.6
Any Way You Slice It
8.7
Surface Area
9
Circles and the Pythagorean Theorem
9.1
The Theorem of Pythagoras
9.2
The Converse of the Pythagorean Theorem
9.3
Radical Expressions
9.4
Story Problems
9.5
Distance in Coordinate Geometry
9.6
Circles and the Pythagorean Theorem
10
Volume
10.1
The Geometry of Solids
10.2
Volume of Prisms and Cylinders
10.3
Volume of Pyramids and Cones
10.4
Volume Problems
10.5
Displacement and Density
10.6
Volume of a Sphere
10.7
Surface Area of a Sphere
11
Similarity
11.1
Proportion and Reasoning
11.2
Similar Triangles
11.3
Indirect Measurement with Similar Triangles
11.4
Corresponding Parts of Similar Triangles
11.5
Proportions with Area
11.6
Proportions with Volume
11.7
Proportional Segments Between Parallel Lines
12
Transforming Functions
12.1
Trigonometric Ratios
12.2
Problem Solving with Right Triangles
12.3
The Law of Sines
12.4
The Law of Cosines
12.5
Problem Solving with Trigonometry
13
Geometry as a Mathematical System
13.1
The Premises of Geometry
13.2
Planning a Geometry Proof
13.3
Triangle Proofs
13.4
Quadrilateral Proofs
13.5
Indirect Proof
13.6
Circle Proofs
13.7
Similarity Proofs
13.8
Coordinate Proof
Textbook Solutions for Discovering Geometry: An Investigative Approach
Chapter 9.5 Problem 11
Question
Mini-Investigation What is the equation of a circle with radius r and center ( h, k )? Use graph paper and your compass, or geometry software, to investigate and make a conjecture. a. Given its center and radius, graph each circle. Circle A: center = ( 1, 2 ) , r = 8 Circle B: center = ( 0, 2 ) , r = 6 b. On each circle, select any point and label it ( x, y ). Use the distance formula to write an equation for the distance from the center of the circle to ( x, y ). c. Look for patterns, then copy and complete the conjecture. Conjecture: The equation of a circle with radius r and center (h, k) is (x ) 2 + (y ) 2 = ( ) 2 ( Equation of a Circle )
Solution
The first step in solving 9.5 problem number 11 trying to solve the problem we have to refer to the textbook question: Mini-Investigation What is the equation of a circle with radius r and center ( h, k )? Use graph paper and your compass, or geometry software, to investigate and make a conjecture. a. Given its center and radius, graph each circle. Circle A: center = ( 1, 2 ) , r = 8 Circle B: center = ( 0, 2 ) , r = 6 b. On each circle, select any point and label it ( x, y ). Use the distance formula to write an equation for the distance from the center of the circle to ( x, y ). c. Look for patterns, then copy and complete the conjecture. Conjecture: The equation of a circle with radius r and center (h, k) is (x ) 2 + (y ) 2 = ( ) 2 ( Equation of a Circle )
From the textbook chapter Distance in Coordinate Geometry you will find a few key concepts needed to solve this.
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Title
Discovering Geometry: An Investigative Approach 4
Author
Michael Serra
ISBN
9781559538824
Mini-Investigation What is the equation of a circle with radius r and center ( h, k )
Chapter 9.5 textbook questions
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Chapter 9: Problem 1 Discovering Geometry: An Investigative Approach 4
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Chapter 9: Problem 2 Discovering Geometry: An Investigative Approach 4
In Exercises 13, find the distance between each pair of points. ( 15, 37 ), ( 42, 73 )
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Chapter 9: Problem 3 Discovering Geometry: An Investigative Approach 4
In Exercises 13, find the distance between each pair of points. ( 19, 16 ), ( 3, 14 )
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Chapter 9: Problem 4 Discovering Geometry: An Investigative Approach 4
Look back at the diagram of Vikis and Scotts locations on page 502. Assume each block is approximately 50 meters long. What is the shortest distance, to the nearest meter, from Viki to Scott?
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Chapter 9: Problem 5 Discovering Geometry: An Investigative Approach 4
Find the perimeter of ABC with vertices A( 2, 4 ), B( 8, 12 ), and C( 24, 0 ).
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Chapter 9: Problem 6 Discovering Geometry: An Investigative Approach 4
Determine whether DEF with vertices D( 6, 6 ), E( 39, 12 ), and F( 24, 18 ) is scalene, isosceles, or equilateral.
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Chapter 9: Problem 7 Discovering Geometry: An Investigative Approach 4
For Exercises 710, graph each quadrilateral using the given vertices. Then use the distance formula and the slope formula to determine the most specific name for each quadrilateral: trapezoid, kite, rectangle, rhombus, square, parallelogram, or just quadrilateral. A( 6, 8 ), B( 9, 7 ), C( 7, 1 ), D( 4, 2 )
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Chapter 9: Problem 8 Discovering Geometry: An Investigative Approach 4
For Exercises 710, graph each quadrilateral using the given vertices. Then use the distance formula and the slope formula to determine the most specific name for each quadrilateral: trapezoid, kite, rectangle, rhombus, square, parallelogram, or just quadrilateral. E( 1, 2 ), F( 5, 5 ), G( 2, 8 ), H( 2, 5 )
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Chapter 9: Problem 9 Discovering Geometry: An Investigative Approach 4
For Exercises 710, graph each quadrilateral using the given vertices. Then use the distance formula and the slope formula to determine the most specific name for each quadrilateral: trapezoid, kite, rectangle, rhombus, square, parallelogram, or just quadrilateral. I( 4, 0 ), J( 7, 1 ), K( 8, 2 ), L( 4, 5 )
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Chapter 9: Problem 10 Discovering Geometry: An Investigative Approach 4
For Exercises 710, graph each quadrilateral using the given vertices. Then use the distance formula and the slope formula to determine the most specific name for each quadrilateral: trapezoid, kite, rectangle, rhombus, square, parallelogram, or just quadrilateral. M( 3, 5 ), N( 1, 1 ), O( 3, 3 ), P( 1, 7 )
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Chapter 9: Problem 11 Discovering Geometry: An Investigative Approach 4
Mini-Investigation What is the equation of a circle with radius r and center ( h, k )? Use graph paper and your compass, or geometry software, to investigate and make a conjecture. a. Given its center and radius, graph each circle. Circle A: center = ( 1, 2 ) , r = 8 Circle B: center = ( 0, 2 ) , r = 6 b. On each circle, select any point and label it ( x, y ). Use the distance formula to write an equation for the distance from the center of the circle to ( x, y ). c. Look for patterns, then copy and complete the conjecture. Conjecture: The equation of a circle with radius r and center (h, k) is (x ) 2 + (y ) 2 = ( ) 2 ( Equation of a Circle )
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Chapter 9: Problem 12 Discovering Geometry: An Investigative Approach 4
Find the equation of the circle with center ( 2, 0 ) and radius 5.
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Chapter 9: Problem 13 Discovering Geometry: An Investigative Approach 4
Find the radius and center of the circle x2 + (y 1)2 = 81.
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Chapter 9: Problem 14 Discovering Geometry: An Investigative Approach 4
The center of a circle is ( 3, 1 ). One point on the circle is ( 6, 2 ). Find the equationof the circle.
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Chapter 9: Problem 15 Discovering Geometry: An Investigative Approach 4
Mini-Investigation How would you find the distance between two points in a three-dimensional coordinate system? Investigate and make a conjecture. a. What is the distance from the origin ( 0, 0, 0 ) to ( 2, 1, 3 ) ? b. What is the distance between P( 1, 2, 3 ) and Q( 5, 6, 15 ) ? c. Use your observations to make a conjecture. Conjecture: If A(x1 , y1 , z1) and B(x2 , y2 , z2) are two points in a three-dimensional coordinate system, then the distance AB is ( Three-dimensional Distance Formula)
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Chapter 9: Problem 16 Discovering Geometry: An Investigative Approach 4
Find the distance between the points ( 12, 9, 13 ) and (28, 75, 52).
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Chapter 9: Problem 17 Discovering Geometry: An Investigative Approach 4
Find the longest possible diagonal in the prism at right.
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Chapter 9: Problem 18 Discovering Geometry: An Investigative Approach 4
Find the rule for this number pattern: 1 3 3 = 4 0 2 4 3 = 5 1 3 5 3 = 6 2 4 6 3 = 7 3 5 7 3 = 8 4
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Chapter 9: Problem 19 Discovering Geometry: An Investigative Approach 4
Find the coordinates of A .
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Chapter 9: Problem 21 Discovering Geometry: An Investigative Approach 4
The large triangle is equilateral. Find x and y
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Chapter 9: Problem 22 Discovering Geometry: An Investigative Approach 4
Antonio is a biologist studying life in a pond. He needs to know how deep the water is. He notices a water lily sticking straight up from the water, whose blossom is 8 cm above the waters surface. Antonio pulls the lily to one side, keeping the stem straight, until the blossom touches the water at a spot 40 cm from where the stem first broke the waters surface. How is Antonio able to calculate the depth of the water? What is the depth?
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Chapter 9: Problem 23 Discovering Geometry: An Investigative Approach 4
C U R T is the image of CURT under a rotation transformation. Copy the polygon and its image onto patty paper. Find the center of rotation and the measure of the angle of rotation. Explain your method.
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Chapter 9: Problem 24 Discovering Geometry: An Investigative Approach 4
Developing Proof Explain why the opposite sides of a regular hexagon are parallel.
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