Geometry 7-6 Circles, Arcs, Circumference and Arc Length

Review

Areas

Area Area of a Triangle

Theorem The Pythagorean theorem In a right triangle, the sum of the squares of the legs of the triangle equals the square of the hypotenuse of the triangle A C b B a c

Theorem Converse of the Pythagorean theorem If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle. A C b B a c

Converse of Pythagorean

Theorem 45° – 45° – 90° Triangle In a 45° – 45° – 90° triangle the hypotenuse is the square root of two times as long as each leg

Theorem 30° – 60° – 90° Triangle In a 30° – 60° – 90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is the square root of three times as long as the shorter leg

Area

Vocabulary

Area

Definitions - Review

Circle Vocabulary (review) Circle – The set of all points in a plane that are equidistant from a given point Center – Equidistant point of a circle Radius – Distance from the center of a circle to a point on the circle Diameter – Distance from a point on the circle to another point on the circle through the center of the circle Congruent Circles – Circles with congruent radii Central Angle – Angle with vertex at the center of the circle

Vocabluary New Definitions

Arc Vocabulary Part of the circle that measures between 180° and 360° Major Arc

Arc Vocabulary Part of the circle that measures between 0° and 180° Minor Arc

Arc Vocabulary An arc whose endpoints are the endpoints of a diameter of a circle Semicircle

Arc Vocabulary The measure of the arcs central angle Measure of a Minor Arc The difference between 360° and the measure of its associated minor arc Measure of a Major Arc

Arc Vocabulary Intercepted Arc Arc within an angle

Arc Vocabulary (Summary) An angle whose vertex is the center of a circle Central Angle Part of the circle that measures between 180° and 360° Major Arc Minor Arc Part of the circle that measures between 0° and 180° Semicircle An arc whose endpoints are the endpoints of a diameter of a circle Measure of a Minor Arc Measure of a Major Arc The measure of the arcs central angle The difference between 360° and the measure of its associated minor arc

New Theorem

Example

Three letters required for major arcs

Example

Arc Length What fraction of a circle is the arc? 1/4=90°/360°

Arc Length What fraction of a circle is the arc? 1/2=180°/360°

Arc Length What fraction of a circle is the arc? 1/3=120°/360°

Arc Length The measure of an arc is calculated in units of degrees, but arc length is calculated in units of distance

Investigation For circles T, O and P, calculate the following in your notes.

Investigation Find the circumference for each circle.

Investigation Find the Arc Length for each circle.

Investigation What is the formula for arc length of a circle?

Arc Length Theorem The length of an arc equals the fraction of the arc to the circle times the circumference Arc Length = Fraction * Circumference

Example

Arc Length - Example

Practice

Arc Length - Practice

Practice

Minor Arc

Semicircle

Major Arc

Minor Arc

Semicircle

Minor Arc

70°

110°

180°

210°

70°

290°

280°

130°

165°

60°

Arc Length - Practice

Practice

Homework Pages 390 – – 8, 16 – 24 even, 28 – 38 even, 42, 44, 52, 63, 64, 68,