APS Geometry Unit 5 Book

Wendy Sanchez, May 9, 2019

MGSE9-12.G.C.1 Understand that all circles are similar. MGSE9-12.G.C.2 Identify and describe relationships among inscribed angles, radii, chords, tangents, and secants. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. MGSE9-12.G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. MGSE9-12.G.C.4 Construct a tangent line from a point outside a given circle to the circle. MGSE9-12.G.C.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. MGSE9-12.G.GMD.1 Give informal arguments for geometric formulas. a. Give informal arguments for the formulas of the circumference of a circle and area of a circle using dissection arguments and informal limit arguments. b. Give informal arguments for the formula of the volume of a cylinder, pyramid, and cone using Cavalieri’s principle. MGSE9-12.G.GMD.2 Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. MGSE9-12.G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Visualize relationships between two-dimensional and three-dimensional objects MGSE9-12.G.GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects generated by rotations of two-dimensional objects.

Table of Contents

  1. MGSE9-12.G.C.1
    1. Similar circles 1 APS
    2. Similar Circles 2 APS
    3. Proving Circles Similar by APS
    4. Similar Circles? APS
  2. MGSE9-12.G.C.2
    1. Exploring Inscribed & Circumscribed Angles APS
    2. Practice solving for arcs and angles in a circle APS
  3. MGSE9-12.G.C.3
    1. Construction: Circumscribed and Inscribed Circles APS
    2. Angle properties of a quadrilateral inscribed in a circle APS
    3. Opposite Angles of a Quadrilateral Inscribed in a Circle APS
    4. Inscribed Quadrilateral's Angles Relationships APS
  4. MGSE9-12.G.C.4
    1. Construction: Tangents to a Circle from a Point Outside the Circle APS
    2. Exploration: Tangent to a point outside a circle APS
  5. MGSE9-12.G.C.5
    1. Exploration: Arc Length and Sector Area APS
    2. Investigate Area of a Sector and Arc Length APS
  6. MGSE9-12.G.GMD.1
    1. Exploration: Circumference and Area of a Circle APS
  7. MGSE9-12.G.GMD.2
  8. MGSE9-12.G.GMD.3
    1. Sections of Rectangular Prisms (Cuboids) APS
    2. Sections of Triangular Prisms APS
    3. Sections of Rectangular Pyramids APS
    4. Pyramid section APS
    5. Sections of Triangular Pyramids APS
    6. Sections of Cylinders APS
    7. Sections of Cones APS
    8. Section of cone APS
    9. Sections of Spheres APS
    10. Section of sphere 2 APS
    11. Exploring Sections of Cubes APS
    12. Cutting a Sphere APS
  9. MGSE9-12.G.GMD.4

1.

MGSE9-12.G.C.1

  • 1. Similar circles 1 APS
  • 2. Similar Circles 2 APS
  • 3. Proving Circles Similar by APS
  • 4. Similar Circles? APS

1.1.

Similar circles 1 APS

1. Drag the sliders up and down to make the pre-image and image circles different sizes. How does the scale factor change as you move the sliders?
2. Use the dilate button to enlarge or reduce the pre-image with the given scale factor. Describe the dilated pre-image.
Use the translate button to slide the dilated pre-image over to the image. Does the new circle map onto the image?
Explain how all circles are similar.
– S Sharma, Ken Garrabrant

1.2.

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2.

MGSE9-12.G.C.2

  • 1. Exploring Inscribed & Circumscribed Angles APS
  • 2. Practice solving for arcs and angles in a circle APS

2.1.

Exploring Inscribed & Circumscribed Angles APS

– badboy_wit03

2.2.

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3.

MGSE9-12.G.C.3

  • 1. Construction: Circumscribed and Inscribed Circles APS
  • 2. Angle properties of a quadrilateral inscribed in a circle APS
  • 3. Opposite Angles of a Quadrilateral Inscribed in a Circle APS
  • 4. Inscribed Quadrilateral's Angles Relationships APS

3.1.

Construction: Circumscribed and Inscribed Circles APS

Construction-1: Circumscribed Circle
Construct a circumscribed circle to the given triangle. Take a screen shot of your final image and save it into google drive.
Construction 2: Inscribed Circle
Construct an inscribed circle to the given triangle. Take a screen shot of your final image and save it into google drive.
– S Sharma, langleyd

3.2.

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4.

MGSE9-12.G.C.4

  • 1. Construction: Tangents to a Circle from a Point Outside the Circle APS
  • 2. Exploration: Tangent to a point outside a circle APS

4.1.

Construction: Tangents to a Circle from a Point Outside the Circle APS

This applet shows how to construct tangents to a circle from a point outside the circle.
1. What happens when the point A is inside the circle ?[br]2. What happens when the point A coincides with point M ?
– S Sharma, Sanjay Gulati

4.2.

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5.

MGSE9-12.G.C.5

  • 1. Exploration: Arc Length and Sector Area APS
  • 2. Investigate Area of a Sector and Arc Length APS

5.1.

Exploration: Arc Length and Sector Area APS

– S Sharma, Rasha Tarek

5.2.

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6.

MGSE9-12.G.GMD.1

  • 1. Exploration: Circumference and Area of a Circle APS

6.1.

Exploration: Circumference and Area of a Circle APS

This applet illustrates formulas for the circumference by rolling a circle along a line, then measuring the length in units of the diameter, or the radius.[br][br]Check the "Show rolling circle" box, then roll the circle with the slider labeled "Roll circle".[br]Check "Show C measured with r, d."[br][br]This leads to a formula for the area of a circle, too. Cut it apart into sectors, and rearrange into a "parallelogram".[br][br]Check the last three boxes to see this.
Formulas:[br][br]Circumference = [math]\pi[/math] diameter (that is, [math]C=\pi d[/math])[br]Circumference = 2[math]\cdot\pi\cdot[/math]radius (that is, [math]C=2\pi r[/math])[br][br]Area = Circumference [math]\cdot[/math] radius [br]Area = [math]\pi\cdot[/math] radius squared (that is, [math]A=\pi r^2[/math])
– S Sharma, Susan Addington

7.

MGSE9-12.G.GMD.2


    This chapter does not contain any resources yet.

8.

MGSE9-12.G.GMD.3

Use these worksheets to gain experience with the cross sections and solids

  • 1. Sections of Rectangular Prisms (Cuboids) APS
  • 2. Sections of Triangular Prisms APS
  • 3. Sections of Rectangular Pyramids APS
  • 4. Pyramid section APS
  • 5. Sections of Triangular Pyramids APS
  • 6. Sections of Cylinders APS
  • 7. Sections of Cones APS
  • 8. Section of cone APS
  • 9. Sections of Spheres APS
  • 10. Section of sphere 2 APS
  • 11. Exploring Sections of Cubes APS
  • 12. Cutting a Sphere APS

8.1.

Sections of Rectangular Prisms (Cuboids) APS

Drag the blue points to see the different sections of the rectangular prism (cuboid).
Anthony Or. GeoGebra Institute of Hong Kong
– S Sharma, Anthony OR 柯志明

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8.4.

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8.5.

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8.6.

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8.7.

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8.8.

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8.9.

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8.10.

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8.11.

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8.12.

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9.

MGSE9-12.G.GMD.4


    This chapter does not contain any resources yet.

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