THEOREMS INVOLVING CHORDS, ARCS, CENTRAL ANGLES || GRADE 10 MATHEMATICS Q2

Summary

In this video lesson, key theorems related to chords, arcs, and central angles in circles are explored. It covers the relationships between minor arcs and their corresponding central angles and chords, emphasizing congruence and equidistance from the center. The lesson also explains the arc addition postulate, the concept of sectors and segments in a circle, and how to calculate their areas. Additionally, it introduces the method for determining arc length using proportions, providing practical examples throughout to enhance understanding.

Takeaways

• 😀 In a circle, two minor arcs are congruent if their corresponding central angles are congruent.
• 😀 Minor arcs are congruent if and only if their corresponding chords are congruent.
• 😀 A diameter bisects an arc with the same endpoints if it is perpendicular to the chord.
• 😀 Chords that are equidistant from the center of a circle are congruent.
• 😀 Any two congruent arcs are equidistant from the center of the circle.
• 😀 The arc addition postulate states that the measure of an arc formed by two adjacent arcs is the sum of their measures.
• 😀 A sector of a circle is defined as the region bounded by an arc and the two radii connecting the endpoints of the arc.
• 😀 To find the area of a sector, multiply the ratio of the arc's measure over 360 by the area of the entire circle.
• 😀 The area of a segment of a circle can be found by subtracting the area of the triangle formed by the endpoints of the arc from the area of the sector.
• 😀 The arc length can be calculated using the proportion of the arc's degree measure to 360 degrees and the circumference of the circle.

Q & A

What is the first theorem discussed in the video regarding minor arcs and central angles?

The first theorem states that in a circle (or in congruent circles), two minor arcs are congruent if and only if their corresponding central angles are congruent.

How can one determine if two minor arcs are congruent according to the second theorem?

According to the second theorem, two minor arcs are congruent if and only if their corresponding chords are congruent.

What does the third theorem about diameters and arcs state?

The third theorem states that a diameter bisects an arc with the same endpoints if and only if it is perpendicular to the chord.

What is the significance of equidistance in relation to chords as stated in the fourth theorem?

The fourth theorem states that in the same circle (or in congruent circles), chords that are equidistant from the center are congruent.

What does the fifth theorem say about congruent arcs and their distance from the center?

The fifth theorem states that in the same circle (or in congruent circles), any two congruent arcs are equidistant from the center.

Explain the Arc Addition Postulate mentioned in the video.

The Arc Addition Postulate states that the measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

How is the area of a sector calculated?

The area of a sector is calculated using the formula: Area of Sector = (Measure of Arc / 360) * Area of Circle.

What is a segment of a circle, as described in the video?

A segment of a circle is the region bounded by an arc and the segment joining its endpoints.

How is arc length determined using proportions?

Arc length can be determined using the proportion: A / 360 = L / (2πr), where A is the degree measure of the arc, L is the arc length, and r is the radius of the circle.

What example is given to demonstrate how to find the area of a sector?

An example is provided where the radius is 4 feet, and the measure of the arc is 80 degrees, resulting in an area of the sector being 32/9 square feet.