Curriculum placement
Geometry. This describes the skill, not a learner estimate.
Why every circle is similar to every other
Prove circle similarity using translations and dilation
Problem types
- Map one circle onto another by a dilation
- Build the proof that two circles are similar
- Justify or find the flaw in a circle-similarity argument
- The transformation that maps any circle onto any other
Builds on
- Proving triangles similar
- How a dilation changes a figure · Useful idea
A dilation maps one circle onto any other, so all circles are similar.
- Constructing the image of a figure · Useful idea
Mapping one circle onto another uses a translation followed by a dilation, which is a composition chosen like any other figure-to-image sequence.
Leads to
- Arc length and sector area · Extend the idea
Because circles are similar, arc length and sector area are fixed fractions of circumference and area, scaling with the radius.
- Equations of circles · See it applied
Every circle is a dilation of the unit circle, so x squared + y squared = r squared is the unit circle scaled by r.
- Radians and the unit circle · Extend the idea
All circles are similar, so arc length divided by radius is the same for a given central angle, and that ratio is the angle in radians.
Related skills
- Sine, cosine, and tangent · Shared foundations
- Theorems about similar triangles · Shared foundations
Open the interactive map to explore lessons and skill connections.