Definition: A central angle is an angle whose vertex is the center O of a circle and whose legs are radii intersecting the circle in two different points on the circumference of the circle. Show The central angle is also known as the arc’s angular distance. Central Angle of a CircleCharacteristicsA central angle is an angle formed by two radii with the vertex at the center of the circle.
UsesThe central angle theorem is very useful in solving questions that deals with angles within circles. By using the central angle theorem, many complicated circle questions can be simplified into a simple one. The central angle theorem is central to many circle questions. ImportanceCentral angle is equal to the arc length. Therefore, to find arc length and area of the sector, central angle plays the most important role. If we know the central angle, we can find both arc length and area of sector. Arc length is the portion of a circumference of a circle made by the central angle. Area of sector is the region bounded by the two radii and the arc length lie between these two radii. Central Angle FormulaThe central angle formula is calculated using the arc measure and inscribed angle of a circle. The angle measure of the central angle is congruent to the measure of the intercepted arc. Central angle = Intercepted arc The measure of central angle is twice the measure of the inscribed angle subtended by the same arc. Central angle = 2 (inscribed angle) Central angle (in radians) = (Arc Length)/Radius Central angle (in radians) = (2 (area of sector))/〖(Radius)〗^2 Central Angle TheoremsCircle theorems involving Central Angle
Calculating Central Angle and Intercepted ArcHere’s how to find the central angle of a circle. Intercepted arc formula
Example 1Determine the value of a in the circle shown below. Solution: The central angle = intercepted arc 60° = (4a + 8)° Simplify: 4a + 8 = 60 Subtract 8 both sides 4a = 52 Divide both sides by 4 a = 13 Example 2Determine the measure of ∠AOB in the circle shown below. Solution: ∠AOB is a central angle and ∠ACB is an inscribed angle subtended on the same arc AB. We know that the central angle is twice the inscribed angle if they are subtended on the same arc. Therefore, ∠AOB = 2∠ACB ∠AOB = 2(64°) ∠AOB = 128° Example 3Determine the measure of arc DC in the circle shown below. Solution: We know that In a circle, congruent central angles have congruent arcs. In the given circle, central angles are congruent ∠AOB ≅ ∠COD Therefore their intercepted arc should have to be congruent. (AB) ̂ = (CD) ̂ (AB) ̂ = 45° (given) (CD) ̂ = 45° Example 4Determine the value of x, if the ∠AOB ≅ ∠COD, and chord AB = 6. Solution: In a circle congruent central angles have congruent chords. In the given circle, ∠AOB and ∠CODare central angles It is given that ∠AOB ≅ ∠COD. Therefore, Chord AB ≅ Chord CD Chord AB = 6 (given) Chord CD = 6. Given f(x) = 3x^2 - 2x + 1 and g(x) = x - 4, find (f*g)(x) Application Do the following: A. Cite at least two situation that illustrates arithmetic and geometric sequences (one for each), and explain why. B. G … Vegetable Garden In the middle of a crisis where establishments are closed and prime commodities are hard to find, we think of having our own vegetabl … 3x-8=x² standard form. 2 x² = 36? what is the anwer? list the elements of the following set T = {x l x is a city in palawan} what is the correct factors of 9x² + 12xy +4y²? pake answer po sa math DESCRIBED SET A IN VEN DIAGRAM GRADE 7 VII. ACTION Direction: In your barangay identify and site anything that uses sets. Give at least two importance of it. Write your answer in the space … |