Circles, II Chords Arcs.

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Presentation transcript:

Circles, II Chords Arcs

Definition – Arc A central angle separates the circle into two parts, each of which is an arc. The measure of each arc is related to the measure of its central angle.

Arcs

Arc Addition Postulate The measure of an arc formed by two adjacent (neighboring) arcs is the sum of the measures of the two arcs.

Example 1 – Measures of Arcs Compute the measurement of arc BE

Example 1 – Measures of Arcs Compute the measurement of arc BE

Example 1 – Measures of Arcs Compute the measurement of arc BE

Example 1 – Measures of Arcs Compute the measurement of arc BE

Example 2 – Measures of Arcs Compute the measurement of arc CE

Example 2 – Measures of Arcs Compute the measurement of arc CE Arc CE is a minor arc

Example 2 – Measures of Arcs Compute the measurement of arc CE Arc CE is a minor arc 𝐶𝐸 = 𝐶𝐵 + 𝐵𝐸

Example 2 – Measures of Arcs Compute the measurement of arc CE Arc CE is a minor arc 𝐶𝐸 = 𝐶𝐵 + 𝐵𝐸 𝐶𝐸 =90+50=140

Example 3 – Measures of Arcs Compute the measurement of arc ACE

Example 3 – Measures of Arcs Compute the measurement of arc ACE Arc ACE is a major arc

Example 3 – Measures of Arcs Compute the measurement of arc ACE Arc ACE is a major arc 𝐴𝐶𝐸 = 𝐴𝐷 + 𝐷𝐵𝐸

Example 3 – Measures of Arcs Compute the measurement of arc ACE Arc ACE is a major arc 𝐴𝐶𝐸 = 𝐴𝐷 + 𝐷𝐵𝐸 𝐴𝐶𝐸 =50+180=230

CW – Arcs

Arcs and Chords – 2 Theorems In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

Arcs and Chords – 2 Theorems In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

Arcs and Chords – 2 Theorems In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. 𝐴𝐵 =80

Arcs and Chords – 2 Theorems In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. 𝐴𝐵 =80 𝐴𝐵 =80= 𝐶𝐷

Arcs and Chords – 2 Theorems In a circle, if a diameter (or radius) is perpendicular to a chord, then it bisects (cut in half) the chord and its arc.

Arcs and Chords – 2 Theorems In a circle, if a diameter (or radius) is perpendicular to a chord, then it bisects (cut in half) the chord and its arc.

Example 1 Circle O has a radius of 13 inches. Radius OB is perpendicular to chord CD, which is 24 inches long.

Example 1 Circle O has a radius of 13 inches. Radius OB is perpendicular to chord CD, which is 24 inches long.

Example 1 Circle O has a radius of 13 inches. Radius OB is perpendicular to chord CD, which is 24 inches long.

Example 1 Circle O has a radius of 13 inches. Radius OB is perpendicular to chord CD, which is 24 inches long.

Example 2 Circle O has a radius of 13 inches. Radius OB is perpendicular to chord CD, which is 24 inches long. Compute length of segment OX.

Example 2 Circle O has a radius of 13 inches. Radius OB is perpendicular to chord CD, which is 24 inches long. Compute length of segment OX. Draw segment CO, CO =

Example 2 Circle O has a radius of 13 inches. Radius OB is perpendicular to chord CD, which is 24 inches long. Compute length of segment OX. Draw segment CO, CO = 13

Example 2 Circle O has a radius of 13 inches. Radius OB is perpendicular to chord CD, which is 24 inches long. Compute length of segment OX. Draw segment CO, CO = 13 CX = 12, by the second theorem.

Example 2 Circle O has a radius of 13 inches. Radius OB is perpendicular to chord CD, which is 24 inches long. Compute length of segment OX. Draw segment CO, CO = 13 CX = 12, by the second theorem.

Example 2 Circle O has a radius of 13 inches. Radius OB is perpendicular to chord CD, which is 24 inches long. Compute length of segment OX. Draw segment CO, CO = 13 CX = 12, by the second theorem.

Example 2 Circle O has a radius of 13 inches. Radius OB is perpendicular to chord CD, which is 24 inches long. Compute length of segment OX. Draw segment CO, CO = 13 CX = 12, by the second theorem.

Example 3 Circle W has a radius of 10 cm. Radius WL is perpendicular to chord HK, which is 16 cm long. Compute 𝐽𝐿

Example 3 Circle W has a radius of 10 cm. Radius WL is perpendicular to chord HK, which is 16 cm long. Compute 𝐽𝐿 Draw 𝑊𝐾

Example 3 Circle W has a radius of 10 cm. Radius WL is perpendicular to chord HK, which is 16 cm long. Compute 𝐽𝐿 Draw 𝑊𝐾

Example 3 Circle W has a radius of 10 cm. Radius WL is perpendicular to chord HK, which is 16 cm long. Compute 𝐽𝐿 Draw 𝑊𝐾

Example 3 Circle W has a radius of 10 cm. Radius WL is perpendicular to chord HK, which is 16 cm long. Compute 𝐽𝐿 Draw 𝑊𝐾

Example 3 Circle W has a radius of 10 cm. Radius WL is perpendicular to chord HK, which is 16 cm long. Compute 𝐽𝐿 Draw 𝑊𝐾

Example 3 Circle W has a radius of 10 cm. Radius WL is perpendicular to chord HK, which is 16 cm long. Compute 𝐽𝐿 Draw 𝑊𝐾

Example 3 Circle W has a radius of 10 cm. Radius WL is perpendicular to chord HK, which is 16 cm long. Compute 𝐽𝐿 Draw 𝑊𝐾

Example 3 Circle W has a radius of 10 cm. Radius WL is perpendicular to chord HK, which is 16 cm long. Compute 𝐽𝐿 Draw 𝑊𝐾

CW 2, HW