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.

Content Standard(s):

Mathematics MA2015 (2016) Grade: 9-12 Geometry

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

Mathematics MA2019 (2019) Grade: 9-12 Geometry with Data Analysis

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

Unpacked Content

Evidence Of Student Attainment:

Students:

Given an arc intercepted by an angle,

Use dilations to create arcs intercepted by the same central angle with radii of various sizes (including using dynamic geometry software), and use the ratios of the arc lengths and radii to make conjectures regarding possible relationship between the arc length and the radius.

Justify the conjecture for the formula for any arc length (i.e., since 2πr is the circumference of the whole circle, a piece of the circle is reduced by the ratio of the arc angle to a full angle (360)).

Find the ratio of the arc length to the radius of each intercepted arc and use the ratio to name the angle calling this the radian measure of the angle by extending the definition of one radian as the angle which intercepts an arc of the same length as the radius.

Develop the formula for the area of a sector by interpreting a circle as a complete revolution and a sector as a fractional part of a revolution.

Teacher Vocabulary:

Similarity

Constant of proportionality

Sector

Arc

Derive

Arc length

Radian measure

Area of sector

Central angle

Dilation

Knowledge:

Students know:

Techniques to use dilations (including using dynamic geometry software) to create circles with arcs intercepted by same central angles.

Techniques to find arc length.

Formulas for area and circumference of a circle.

Skills:

Students are able to:

Reason from progressive examples using dynamic geometry software to form conjectures about relationships among arc length, central angles, and the radius.

Use logical reasoning to justify (or deny) these conjectures and critique the reasoning presented by others.

Interpret a sector as a portion of a circle, and use the ratio of the portion to the whole circle to create a formula for the area of a sector.

Understanding:

Students understand that:

Radians measure the ratio of the arc length to the radius for an intercepted arc.

The ratio of the area of a sector to the area of a circle is proportional to the ratio of the central angle to a complete revolution.

Diverse Learning Needs:

Essential Skills:

Learning Objectives: GEO.20.1: Define arc length, radian measure, and sector.
GEO.20.2: Prove the length of the arc intercepted by an angle is proportional to the radius by similarity.
GEO.20.3: Prove the formula for the area of the sector.
GEO.20.4: Illustrate an arc of a circle by constructing the radii of a circle.

Prior Knowledge Skills:

Identify parts of a circle.

Recall the meaning of a radius and diameter.

Identify all types of angles.

Recognize the attributes of a circle.

Identify and label parts of a circle.

Define diameter, radius, circumference, area of a circle, and formula.

Mathematics MA2019 (2019) Grade: 9-12 Geometry with Data Analysis

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.

Unpacked Content

Evidence Of Student Attainment:

Students:
Given circles with two points on the circle,

Compare the measures of the angles (with and without technology) formed by creating radii to the given points, creating chords from a third point on the circle to the given points, and creating tangents from a third point outside the circle to the given points, and conjecture about possible relationships among the angles.

Use logical reasoning to justify (or deny) the conjectures (in particular justify that an inscribed angle is one half the central angle cutting off the same arc, and the circumscribed angle cutting off that arc is supplementary to the central angle relating all three).

Given circles with chords from a point on the circle to the endpoints of a diameter,

Find the measure of the angles (with and without technology), conjecture about and explain possible relationships.

Use logical reasoning to justify (or deny) the conjectures (in particular justify that an inscribed angle on a diameter is a right angle).

Given a circle with a tangent and radius intersecting at a point on the circle,

Find the measure of the angle at the intersection point (with and without technology), conjecture about and explain possible relationships.

Use logical reasoning to justify (or deny) the conjectures (in particular justify that the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

Teacher Vocabulary:

Central angles

Inscribed angles

Circumscribed angles

Chord

Circumscribed

Tangent

Perpendicular arc

Knowledge:

Students know:

Definitions and characteristics of central, inscribed, and circumscribed angles in a circle.

Techniques to find measures of angles including using technology (dynamic geometry software).

Skills:

Students are able to:

Explain and justify possible relationships among central, inscribed, and circumscribed angles sharing intersection points on the circle.

Accurately find measures of angles (including using technology (dynamic geometry software)) formed from inscribed angles, radii, chords, central angles, circumscribed angles, and tangents.

Understanding:

Students understand that:

Relationships that exist among inscribed angles, radii, and chords may be used to find the measures of other angles when appropriate conditions are given.

Identifying and justifying relationships exist in geometric figures.

Diverse Learning Needs:

Essential Skills:

Learning Objectives: GEO.37.1: Define inscribed angles, central angles, circumscribed angles, radius, chord, tangent, secant, and diameter.
GEO.37.2: Define inscribed and circumscribed circle of a triangle.
GEO.37.3: Apply knowledge of arcs, angles and chords to solve circle related problems.
GEO.37.4: Determine angle values for all angles formed in the exterior, interior and on the circle.
GEO.37.5: Determine lengths of intersecting chords and secants.
GEO.37.6: Discuss the relationship among inscribed angles, radii, and chords.
GEO.37.7: Illustrate inscribed and circumscribed circles of a triangle and quadrilaterals inscribed in a circle.
GEO.37.8: Illustrate radii, chords, diameters, tangents to curve, central, inscribed, and circumscribed angles.

Prior Knowledge Skills:

Identify parts of a circle.

Recall how to find circumference of a circle.

Recall the meaning of a radius and diameter.

Identify all types of angles.

Recognize the attributes of a circle.

Identify and label parts of a circle.

Define diameter, radius, circumference, area of a circle, and formula.

Alabama Alternate Achievement Standards

AAS Standard: M.G.AAS.10.36 Use geometric shapes to describe real-world objects.