All About Circles

GROUP 1 10-ZARA, Mar 29, 2024

[i]In this Book, Group 1 has prepared applets for the following topics related to Circles: [/i] [color=#b20ea8]1. Parts of a Circle 2. Chord and Arc 3. Central and Inscribed Angle 4. Tangents 5. Secants 6. Power Theorems 7. Circle Relationships [/color] [b]Group Members: [/b] Arcoy, Sofia Eliz Karlymei Benigno, Laneah Cortes, Ronald Kenji Guarino, Jefferson Malapitan, Samuel

Table of Contents

  1. Parts of a Circle (Introduction)
    1. Congruent Radii, Congruent Circles
    2. Radius and Diameter
    3. Semicircle
    4. Minor and Major Arc
  2. Chord and Arc
    1. Chords, Arcs, Central Angles
    2. Segment from Center to Chord
    3. Congruent Chords and the Center of a Circle
  3. Central and Inscribed Angles
    1. Arc Addition Postulate
    2. Inscribed Angles
    3. Angle Inscribed in a Semicircle
    4. Inscribed Quadrilateral in a Circle
  4. Secants
    1. Secants Intersecting Outside the Circle
    2. Secants Intersecting Inside the Circle
    3. Tangent-Secant Intersecting Outside the Circle
    4. Tangents Intersecting Outside the Circle
  5. Tangents
    1. Radius and a Tangent at Point of Tangency
    2. Angle Formed by a Chord and a Tangent
    3. Tangents Intersecting Circle at Common External Point
  6. Power Theorems
    1. Chord-Chord Power Theorem
    2. Secant-Secant Power Theorem
    3. Tangent-Secant Power Theorem
  7. Circle Relationships
    1. Common External Tangent
    2. Common Internal Tangent
    3. Externally Tangent Circles
    4. Internally Tangent Circles

1.

Parts of a Circle (Introduction)

  • 1. Congruent Radii, Congruent Circles
  • 2. Radius and Diameter
  • 3. Semicircle
  • 4. Minor and Major Arc

1.1.

Congruent Radii, Congruent Circles

By Kenji Cortes
Observe the two circles above, along with their radii.
Two Circles are said to be congruent if their radii are congruent. Given in the two figures above, Circle A and B are congruent because they both have the radii of 3 units.
– GROUP 1 10-ZARA

1.2.

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1.3.

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1.4.

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2.

Chord and Arc

  • 1. Chords, Arcs, Central Angles
  • 2. Segment from Center to Chord
  • 3. Congruent Chords and the Center of a Circle

2.1.

Chords, Arcs, Central Angles

by Kenji Cortes
Observe the measure of the arcs, central angles, and chords formed in Circle A above.
In Circle A, these are the following observations you should have: [br][br]1. Congruent central angles intercept congruent arcs, in the same manner that congruent arcs are subtended by congruent central angles. [br]2. Congruent chords have congruent arcs.
– GROUP 1 10-ZARA

2.2.

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2.3.

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3.

Central and Inscribed Angles

  • 1. Arc Addition Postulate
  • 2. Inscribed Angles
  • 3. Angle Inscribed in a Semicircle
  • 4. Inscribed Quadrilateral in a Circle

3.1.

Arc Addition Postulate

by Sofia Arcoy
Observe Circle A, and Arcs BC and CD.
The Arc Addition Postulate states that the measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs. In this case, Arc BC (measuring 55.04 degrees) and Arc CD (measuring 61.28 degrees) create Arc BD, which measures the sum of their angles, (116.32 degrees).
– GROUP 1 10-ZARA

3.2.

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3.3.

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3.4.

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4.

Secants

  • 1. Secants Intersecting Outside the Circle
  • 2. Secants Intersecting Inside the Circle
  • 3. Tangent-Secant Intersecting Outside the Circle
  • 4. Tangents Intersecting Outside the Circle

4.1.

Secants Intersecting Outside the Circle

by Sofia Arcoy
Drag Points D, B, and F to observe the relationship of Arcs FB and EC to Angle CDE.
In the figure above, Secants BD and FD intersect at point D, and they form Angle CDE. As you may have observed, the measure of [b]Angle CDE[/b] is equal to [b]one half[/b] the[b] positive difference[/b] of [b]Arcs BF[/b] and[b] CE[/b].
– GROUP 1 10-ZARA

4.2.

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4.3.

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4.4.

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5.

Tangents

  • 1. Radius and a Tangent at Point of Tangency
  • 2. Angle Formed by a Chord and a Tangent
  • 3. Tangents Intersecting Circle at Common External Point

5.1.

Radius and a Tangent at Point of Tangency

by Sofia Arcoy
Drag Circle A/Point B to identify the measurement of the angle they make at the Point of Tangency.
During your investigation, you should be able to see that the radius of [b]Circle A[/b] intersecting with a [b]Tangent [/b]at the [b]Point of Tangency[/b] are[b] perpendicular[/b] with each other.
– GROUP 1 10-ZARA

5.2.

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5.3.

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6.

Power Theorems

  • 1. Chord-Chord Power Theorem
  • 2. Secant-Secant Power Theorem
  • 3. Tangent-Secant Power Theorem

6.1.

Chord-Chord Power Theorem

by Jefferson Guarino
– GROUP 1 10-ZARA

6.2.

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6.3.

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7.

Circle Relationships

  • 1. Common External Tangent
  • 2. Common Internal Tangent
  • 3. Externally Tangent Circles
  • 4. Internally Tangent Circles

7.1.

Common External Tangent

by Sofia Arcoy
Tangent EF is a line that is tangent to two different circles and it does not cross the line segment that connects the centers of the circles.
– GROUP 1 10-ZARA

7.2.

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7.3.

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7.4.

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