View Standards
**Standard(s): **
[MA2015] GEO (9-12) 26 :

[MA2019] GEO-19 (9-12) 37 :

26 ) Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. [G-C3]

[MA2019] GEO-19 (9-12) 37 :

37. Investigate and apply relationships among inscribed angles, radii, and chords, including but not limited to: the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

Module 5, Topic A leads students first to Thales' theorem (an angle drawn from a diameter of a circle to a point on the circle is sure to be a right angle), then to possible converses of Thales' theorem, and finally to the general inscribed-central angle theorem. Students use this result to solve unknown-angle problems. Through this work, students construct triangles and rectangles inscribed in circles and study their properties.

View Standards
**Standard(s): **
[MA2015] GEO (9-12) 24 :

[MA2019] GEO-19 (9-12) 20 :

24 ) Prove that all circles are similar. [G-C1]

[MA2019] GEO-19 (9-12) 20 :

20. Derive and apply the formula for the length of an arc and the formula for the area of a sector.

[MA2019] GEO-19 (9-12) 37 : 37. Investigate and apply relationships among inscribed angles, radii, and chords, including but not limited to: the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

Module 5, Topic B defines the measure of an arc and establishes results relating to chord lengths and the measures of the arcs they subtend. Students build on their knowledge of circles from Module 2 and prove that all circles are similar. Students develop a formula for arc length in addition to a formula for the area of a sector and practice their skills solving unknown area problems.

26 ) Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. [G-C3]

[MA2019] GEO-19 (9-12) 37 :

37. Investigate and apply relationships among inscribed angles, radii, and chords, including but not limited to: the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

In Module 5, Topic C, students explore geometric relations in diagrams of two secant lines, or a secant and tangent line (possibly even two tangent lines), meeting a point inside or outside of a circle. They establish the secant angle theorems and tangent-secant angle theorems. By drawing auxiliary lines, students also notice similar triangles and thereby discover relationships between lengths of line segments appearing in these diagrams.