ALEX Classroom Resource

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.

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.

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.

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.

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.

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.

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).

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.

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.

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.