Honors Geometry

Brian Sterr, Dec 21, 2022

Resources for my 9th grade Honors Geometry Class

Table of Contents

  1. 3D Geometry
    1. Plane: Three points
    2. Plane: Point and Line
    3. Plane: Two Intersecting Lines
    4. Plane: Two Parallel Lines
    5. Extra Credit 1 - 3D Model
  2. Ratio & Proportion
    1. Exterior Angle Bisector Theorem
    2. Varignon's Theorem
    3. Classwork 54 #2
    4. Weeks & Adkins p.214 #13
    5. Projection of a Point onto a Line
  3. Trigonometry
    1. Unit Circle Trigonometry
    2. Unit Circle: Complementary Angles
    3. Law of Sines
    4. The Ambiguous Case of SSA
  4. Area
    1. Shoelace Formula
    2. Area of a Circle
  5. Constructions
    1. Constructing a Golden Triangle
  6. Surface Area & Volume
    1. Creating a Cone
    2. Frustum of a Cone
    3. Surface Area of a Sphere
    4. Cavalieri's Principle
    5. Volume of a Pyramid - Cavalieri's Principle
    6. Volume of a Regular Pyramid
    7. Archimedes Volume of a Sphere
  7. Homework/Classwork Problems
    1. p.519 #33
    2. Classwork 30 #2

1.

3D Geometry

  • 1. Plane: Three points
  • 2. Plane: Point and Line
  • 3. Plane: Two Intersecting Lines
  • 4. Plane: Two Parallel Lines
  • 5. Extra Credit 1 - 3D Model

1.1.

Plane: Three points

Three Non-Collinear points Determine a Plane
– Brian Sterr

1.2.

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1.3.

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1.4.

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1.5.

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2.

Ratio & Proportion

  • 1. Exterior Angle Bisector Theorem
  • 2. Varignon's Theorem
  • 3. Classwork 54 #2
  • 4. Weeks & Adkins p.214 #13
  • 5. Projection of a Point onto a Line

2.1.

Exterior Angle Bisector Theorem

Theorem
An angle bisector of an exterior angle of a triangle divides the opposite side externally into segments proportional to the adjacent sides.
Setup for Proof
Given: [math]\overline{AD}[/math] bisects the exterior angle of [math]\triangle ABC[/math] at [math]A[/math].[br][br]Prove: [math]\frac{DB}{DC}=\frac{AB}{AC}[/math][br][br]Construction:[br]Construct [math]\overline{BE}[/math] parallel to [math]\overline{AD}[/math] through [math]B[/math], intersecting [math]\overline{AC}[/math] at [math]E[/math]
Explore
Move [math]A[/math] to change the type of triangle for [math]\triangle ABC[/math].
Will the angle bisector always intersect [math]\overleftrightarrow{BC}[/math]?
– Brian Sterr

2.2.

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2.3.

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2.4.

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2.5.

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3.

Trigonometry

  • 1. Unit Circle Trigonometry
  • 2. Unit Circle: Complementary Angles
  • 3. Law of Sines
  • 4. The Ambiguous Case of SSA

3.1.

Unit Circle Trigonometry

Drag the slider for [math]\theta[/math] to adjust the angle. Notice the coordinates of point [math]P[/math] on the unit circle.
– Brian Sterr

3.2.

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3.3.

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3.4.

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4.

Area

  • 1. Shoelace Formula
  • 2. Area of a Circle

4.1.

Shoelace Formula

Triangle Area (Counter-Clockwise)
What if we were to switch points [math]A[/math] and [math]C[/math]?[br]What would happen?
Triangle Area (Clockwise)
Shoelace for Quadrilateral
The formula can also be applied to Quadrilaterals or other polygons in general. Try dragging A,B,C,D around to see what happens if the quadrilateral is not convex, or if it is self-intersecting. Does it seem like the formula is still valid?
– Brian Sterr

4.2.

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5.

Constructions

  • 1. Constructing a Golden Triangle

5.1.

Constructing a Golden Triangle

– Brian Sterr

6.

Surface Area & Volume

  • 1. Creating a Cone
  • 2. Frustum of a Cone
  • 3. Surface Area of a Sphere
  • 4. Cavalieri's Principle
  • 5. Volume of a Pyramid - Cavalieri's Principle
  • 6. Volume of a Regular Pyramid
  • 7. Archimedes Volume of a Sphere

6.1.

Creating a Cone

Creating a Cone
A cone can be created by cutting out a sector of a circle and taping the two radii together.[br]In the diagram of the cone, we can see the Pythagorean relationship between the height [math]h[/math], radius of the base of the cone [math]r[/math], and slant height of the cone [math]l[/math]:[br][br][math]r^2+h^2=l^2[/math]
Central Angle and Radius
If we fix the slant height [math]l[/math] (which corresponds to the radius of the flat sector, what is the radius [math]r[/math], in terms of [math]l[/math]?
Relating the Angles
How does the central angle of the sector [math]\alpha[/math] relate to the angle between the altitude and slant height [math]\beta[/math]?
– Brian Sterr

6.2.

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6.3.

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6.4.

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6.5.

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6.6.

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6.7.

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7.

Homework/Classwork Problems

  • 1. p.519 #33
  • 2. Classwork 30 #2

7.1.

p.519 #33

This demonstrates the fact that, for a trapezoid [math]ABCD[/math], given a point [math]P[/math] in its interior, it is not necessarily true that:[br][math]A_{\triangle APD}+A_{\triangle BPC}=A_{\triangle APB}+A_{\triangle PCD}[/math][br][br]Drag point [math]P[/math] around and notice that the sum of the areas of the two triangles is not always half of the area of the entire trapezoid. Since all four triangles sum to the area of the trapezoid, this means that the two sums of areas of the triangles are equal if and only if each sum is half of the area of the trapezoid.
– Brian Sterr

7.2.

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