# G.11 and *G.11 - Circle Properties

G.11    The student will use angles, arcs, chords, tangents, and secants to

a)  investigate, verify, and apply properties of circles;

b)  solve real-world problems involving properties of circles; and

c)  find arc lengths and areas of sectors in circles.

G.11  The student will solve problems, including practical problems, by applying properties of circles. This will include determining

a)  angle measures formed by intersecting chords, secants, and/or tangents;

b)  lengths of segments formed by intersecting chords, secants, and/or tangents;

c)  arc length;

d)  area of a sector.

Bloom's Level:  Evaluate

Adopted: 2009

### BIG IDEAS

• I can place seats on a ferris wheel for perfect balance, get the winning score in a dart game, and cut a pizza into equal pieces.
• I will apply spatial reasoning and visualization as ways to orient my thinking about the physical world, especially in regards to circles since the world is not made up of straight lines.

### UNDERSTANDING THE STANDARD

• Many relationships exist between and among angles, arcs, secants, chords, and tangents of a circle.
• All circles are similar.
• A chord is part of a secant.
• Real-world applications may be drawn from architecture, art, and construction.

·  All circles are similar.

·  A chord is a line segment that joins any two points on a circle. A chord is a segment of a secant.

·  Arcs can be measured in degrees or in units of length.

·  Applications of the properties of circles may be drawn from architecture, art, and construction.

·  Properties of circles can be verified using deductive reasoning, algebraic, and coordinate methods.

·  Inscribed quadrilaterals have opposite angles that are supplementary.

·  Properties associated with segment lengths can be verified using similar triangles.

·  The ratio of the central angle to 360° is proportional to the ratio of the arc length to the circumference of the circle.

·  The ratio of the central angle to 360° is proportional to the ratio of the area of the sector to the area of the circle.

·  The construction for an inscribed equilateral triangle, square and regular hexagon can be justified using properties of a circle.

### ESSENTIALS

Review Area and Circumference of a Circle

G.11a1  Find the lengths, angle measures, and arc measures associated with central angles

G.11c1  Calculate the area of a sector and length of an arc of a circle, using proportions.

·  G.11d1  Calculate the area of a sector.

·  G.11c1  Calculate the length of an arc of a circle.

G.11b1  Solve real-world problems associated with circles, using properties of angles, lines, and arcs.

G.11a2  Find the lengths, angle measures, and arc measures associated with 2 intersecting chords

G.11a3  Find the lengths, angle measures, and arc measures associated with 2 intersecting secants

G.11a4  Find the lengths, angle measures, and arc measures associated with intersecting secant and tangent
G.11a5  Find the lengths, angle measures, and arc measures associated with 2 intersecting tangents

G.11a6  Find the lengths, angle measures, and arc measures associated with inscribed angles

·  G.11b1  Determine segment lengths associated with:

­  two intersecting chords;

­  two intersecting secants;

­  an intersecting secant and tangent; and

­  two intersecting tangents.

·  G.11a1  Determine angle measures and arc measures associated with

­  two intersecting chords;

­  two intersecting secants;

­  an intersecting secant and tangent;

­  two intersecting tangents; and

­  central and inscribed angles.

G.11b2  Solve real-world problems associated with circles, using properties of angles, lines, and arcs.

G.11a7  Verify properties of circles, using deductive reasoning algebraic, and coordinate methods.

·  G.11abcd1  Solve problems, including practical problems, by applying properties of circles.

### KEY VOCABULARY

point, angle, arc, chord, segment, line, tangent, secant, circle, investigate, verify, apply, solve, properties of circles, arc length, area, sector, length, angle measure, arc measure, diameter, radius/radii, center, minor arc, major arc, semicircle,  intersecting chords, intersecting secants, intersecting tangents, central angle, inscribed angle, inscribed, circumscribed, intercepted arc, point of tangency, Pi, deductive reasoning, algebraic methods, coordinate methods, similar

Updated: Feb 22, 2018