Mathematics Subject Outline
Investigation: Intersecting Chords of a Circle
On completion of the investigation I had learned many new important qualities associated with the intersecting chords within a circle theorems. The three theorems studied in this investigation include: Two chords intersecting externally, two chords intersecting internally and the intersection of a chord and tangent. Each theorem can be used to determine different things (e.g. the tangent-chord theorem can be used to determine the approximate distance from the horizon a person is dependant on their height above the earth’s surface).
The most important thing that I learned is how the theorems are derived as well as how to apply these concepts to real life problems. In each case the theorem is derived using similar triangles. By using the similar triangles formed between chords, or tangents and chords, an equation can be formed in order to determine the relationship between certain lengths on the diagram. An example of this for each theorem is:
Example 1
Two Chords Intersecting Externally
By using similar triangles the connection between chord lengths can be seen.
Triangle ACE is similar to Triangle BDE (equal angles)
Example 2
Two Chords Intersecting Internally
By using similar triangles the connection between chord lengths can be seen.
Example 3
The Intersection of a Chord and Tangent
By using similar triangles the connection between chord length and tangent can be seen.
AB2 = AD × CA
AB =
After closer analysis, it is easy to see how the equations for the theorems are derived. The investigation also taught me that these three theorems can be used to solve problems and can be applied to situations in the real world.
The Earth is approximately 12740 km in diametre and using the equation it is possible to calculate the distance from the horizon an object is depending on how high above the earth’s
On completion of the investigation I had learned many new important qualities associated with the intersecting chords within a circle theorems. The three theorems studied in this investigation include: Two chords intersecting externally, two chords intersecting internally and the intersection of a chord and tangent. Each theorem can be used to determine different things (e.g. the tangent-chord theorem can be used to determine the approximate distance from the horizon a person is dependant on their height above the earth’s surface).
The most important thing that I learned is how the theorems are derived as well as how to apply these concepts to real life problems. In each case the theorem is derived using similar triangles. By using the similar triangles formed between chords, or tangents and chords, an equation can be formed in order to determine the relationship between certain lengths on the diagram. An example of this for each theorem is:
Example 1
Two Chords Intersecting Externally
By using similar triangles the connection between chord lengths can be seen.
Triangle ACE is similar to Triangle BDE (equal angles)
Example 2
Two Chords Intersecting Internally
By using similar triangles the connection between chord lengths can be seen.
Example 3
The Intersection of a Chord and Tangent
By using similar triangles the connection between chord length and tangent can be seen.
AB2 = AD × CA
AB =
After closer analysis, it is easy to see how the equations for the theorems are derived. The investigation also taught me that these three theorems can be used to solve problems and can be applied to situations in the real world.
The Earth is approximately 12740 km in diametre and using the equation it is possible to calculate the distance from the horizon an object is depending on how high above the earth’s