A circle has radius 7. Find its circumference in exact form.
Unit 08 · Lesson 05 · Circles, Arcs & Tangency
Tangents and Radius Relationships
Prove that a tangent is perpendicular to the radius at the point of tangency and use equal tangent segments.
Learning intention
Use accurate diagrams, definitions, measurements, and reasoning to prove that a tangent is perpendicular to the radius at the point of tangency and use equal tangent segments.
Success evidence
- Label the figure and units before calculating.
- Select and justify the governing definition, theorem, relationship, or construction.
- Show enough work that another learner can audit the conclusion.
- Check the result against the diagram, scale, and stated constraints.
Visual model
Connect radii, chords, arcs, angles, tangents, secants, sectors, and coordinate equations in a unified circle model.
C=2\pi rA=\pi r^2s=\frac{\theta}{360^\circ}2\pi r(x-h)^2+(y-k)^2=r^2
Worked reasoning
Process: Identify the angle or segment type, locate its intercepted arc or related products, apply the correct theorem, and confirm the result fits the diagram.
- Read the figure and rewrite all given information with correct notation.
- Identify the exact relationship that connects the known and unknown quantities.
- Substitute, construct, or prove one justified step at a time.
- State the result with units and a sentence explaining what it means.
Misconception check: An inscribed angle is half its intercepted arc, while a central angle has the same measure as its intercepted arc.
Khaemenes Academy · Open-Age Geometry
Tangents and Radius Relationships
Find, construct, prove, or explain each result. Show reasoning and label units.
Name
Date
A circle has radius 7. Find its area in exact form.
In a circle of radius 7, find the arc length for a 60° central angle in exact form.
An intercepted arc measures 60°. Find the inscribed angle.
What angle is formed by a radius and a tangent at the point of tangency?
Write the equation of a circle centered at (3,−2) with radius 5.