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Circles DEFINITIONS, ARCS, AND ANGLES IN CIRCLES Key Ideas The following are the basic circle definitions and relationships: The measure of a central angle equals the measure of the intercepted arc. The measure of an inscribed angle equals the measure of the intercepted arc. The sum of the arc measures around a circle equals 360°. All radii of a circle are congruent, and congruent circles have congruent radii. All circles are similar, and one can be mapped to another with only a translation and a dilation. Definitions A circle is the set of points a fixed distance from a point. The point is the center of the circle, and the fixed distance is the radius. Segments and Lines Definitions in Circles
Circle P with radius, chord, diameter, secant, and tangent Math Fact When we say “on” the circle, we mean points on the circumference of the circle.
Points in the interior of the circle are not on the circle.
Consider the equation of the circle, (x − h)2 + (y − k)2 = r2. It is satisfied only by the coordinates of points that lie on the circumference.
When we talk about the area of a circle, we really mean the area enclosed by the circle. Radii, Congruence, and Similarity Every circle has an infinite number of radii. Figure above shows only one radius. The diameter is always equal to twice the radius. Radius and Diameter Theorems The length of the diameter of a circle equals twice the length of its radius.
We can determine if two circles are congruent by comparing the radii. If the radii are congruent, the circles are congruent. Congruent Circles Theorem Two circles are congruent if their radii or if their diameters are congruent. Since two circles with congruent radii are congruent, we can determine a set of rigid motions that will map one onto the other. A circle has an infinite degree of rotational symmetry (a rotation of any angle will map it onto itself). A reflection about any diameter will also map a circle to itself. Therefore, we only need to translate the centers to map one congruent circle onto another.
In Figure below, ⊙P ≅ ⊙Q. We can map ⊙Q onto ⊙P with the translation . Mapping one congruent circle onto another
We saw from Chapter 8 that similar figures have the same shape but may differ in size. All circles by definition have the same shape. The size of any circle is determined only by its radius. When given any two circles, we can scale one circle in size so that it is congruent to the other by applying a dilation with a scale factor equal to the ratio of the radii. If the center of the dilation is the center of the circle, then the original center and its image are coincident and the circles are concentric. The congruent circles can then be mapped to one another by applying the appropriate translation as explained above. Definition Concentric circles—Circles having the same centers. Similar Circles Theorems All circles are similar. Any circle can be mapped to another circle. First dilate by the ratio of the radii. Then translate along a vector determined by the centers of the circles. Figure below shows ⊙Q being mapped to ⊙P with a different radius. The transformations used are as follows: Dilation of ⊙Q with scale factor equal to about center Q. The new radius, Q′A′, is equal to PB. Translation of ⊙Q′ by . Mapping of one circle to another circle of a different radius Example 1 State a specific sequence of transformations that could be used to demonstrate that circle H is similar to circle K. Solution: One possible transformation that would map the two circles would be a translation that maps H to K, followed by a dilation to match the radii. H needs to be translated 1 unit right and 1 unit down. A scale factor of 2 applied to circle H would make the radii congruent: DK,2 ◦ T1,−1 (circle H). Example 2 Circle P has a radius of 6, and circle Q has a radius of 10. State a sequence of transformations that could be used to demonstrate the two circles are similar. Solution: A translation that maps point P to point Q followed by a dilation centered at Q with a scale factor of , or , would map circle P onto circle Q. Since this represents a similarity transformation, the circles must be similar. (circle P). Arcs, Central Angles, and Inscribed Angles An arc is a continuous section of a circle. It is defined by its endpoints, which are on the circle. We denote an arc with the arc symbol and the endpoints of the arc, such as . Arcs are classified as either major arcs, minor arcs, or semicircles:
Minor arc—an arc spanning less than a semicircle Semicircle—an arc spanning exactly of a circle Major arc—an arc spanning more than a semicircle
When naming a semicircle or major arc, we add an additional point to show which semicircle is being specified.
Figure shows minor arc , semicircle , and major arc . Minor arc, semicircle, and major arc
We can refer to two types of angles within a circle—the central angle and the inscribed angle. Definition Central angle—An angle whose vertex is the center of a circle and whose rays intersect the circle. Definition Inscribed angle—An angle whose vertex is on a circle and whose rays intersect the circle Figure shows central angle ∠APB and inscribed angle ∠CMD. Each central angle and inscribed angle cuts off, or intercepts, an arc on the circle. Central ∠APB intercepts . Inscribed angle ∠CMD intercepts . Central angle ∠APB and inscribed angle ∠CMD
An arc has two different dimensions that we consider—its angle measure and its length. The angle measure of an arc is equal to the central angle that intercepts the arc. In Figure, the measure of is equal to the measure of ∠APB. There is also a relationship between an arc and the inscribed angle. The angle measure of an arc is equal to 2 times the angle measure of the inscribed angle that intercepts the arc. Arc Measure Relationships Arc—Central Angle Measures
The angle measure of an arc equals the measure of the central angle that intercepts the arc.
Arc—Inscribed Angle Measures The angle measure of an arc equals twice the measure of the inscribed angle that intercepts the arc. The measure of an inscribed angle equals half the angle measure of the arc it intercepts. Example 1 In circle P, ∠ACB is an inscribed angle and measures 28°. What is the measure of ? Solution: Example 2 In ⊙P, ∠APB is a central angle and ∠ADB is an inscribed angle. If m∠APB = 110°, find m∠ADB. and then use the arc to find the inscribed angle.
The definition of a central angle requires the vertex to be at the center of the circle. Therefore given any arc on a circle, there is only one possible central angle that intercepts the arc. However, there are an infinite number of possible inscribed angles that intercept an arc. They must all have the same angle measure. Congruent Inscribed Angles Theorem Inscribed angles that intercept congruent arcs are congruent. Arcs intercepted by congruent inscribed angles are congruent. This theorem can help us find measures of inscribed angles that intercept a common arc. We can also use this theorem to help prove that two triangles are similar. Example 3 ∠C and ∠D are inscribed angles in ⊙P. If m∠C = 21°, what is m∠D? Prove △ADQ ~ △BCQ. Solution: ∠D ≅ ∠C because they both intercept . So m∠D = m∠C = 21°. ∠A ≅ ∠B because they both intercept .
Therefore △ADQ ~ △BCQ by the AA postulate. Because all radii in a circle are congruent, be on the lookout for isosceles triangles involving two radii in circle problems. Example 4 = 108°. What is the measure of ∠A? = 108°.
△PBA is isosceles, with Arc Addition Consecutive arcs can be added in the same way as angles and segments to get the sum of the larger arc. Three useful arc-angle relationships that can be used in problems involving arc measures are defined. Circle—Arc Sum Theorem The sum of the measures of consecutive arcs around a circle equals 360°.
Semicircle—Arc Sum Theorem The sum of the measures of consecutive arcs around a semicircle equals 180°. Combining this last theorem with the fact that an inscribed angle measures the measure of its intercepted arc leads to the following theorem. Inscribed Angle—Semicircle Theorem An inscribed right angle always intercepts a semicircle. Example 1 Diameter is drawn in circle and .
Solution:
Example 2 Diameter is drawn in ⊙P. If m∠BPC = 31°, what is m∠ARB?
Solution: Since ∠ARB is an inscribed angle, the strategy is to find the measure of the arc that the angle intercepts. Example 3 Points D, J, and K are located on circle O. The three points divide the circle into arcs that are in a 3 : 4 : 5 ratio. What is the measure of the three arcs? Solution: The sum of the arc measures must equal 360°. We can express the three arcs as 3x, 4x, and 5x. The arc measures are 3(30)°, 4(30)°, and 5(30)° = 90°, 120°, and 150° Example 4 In ⊙O, is a diameter. If RS = 4 and RT = 10, find the length of . Express your answer in simplest radical form. Solution: Since TOS is a diameter, inscribed ∠R measures 90° and △RST is a right triangle. Apply the Pythagorean theorem to find TOS.
Math Fact Quadrilaterals that are inscribed in a circle are called cyclic quadrilaterals. Cyclic quadrilaterals have opposite angles that are supplementary.
This property is easily proven by looking at a sketch of a cyclic quadrilateral. Together, arcs and comprise the entire circle, so
Since the intercepted angles measure the arc measures, m∠ABC + m∠ADC = 180°. So the angles are supplementary. The same justification can be used to show ∠A and ∠C are supplementary.
CONGRUENT, PARALLEL, AND PERPENDICULAR CHORDS Key Ideas The following theorems relate arc measures to parallel, perpendicular, and congruent chords:
If two chords in a circle are congruent, they intercept congruent arcs. If two chords in a circle are parallel, the arcs between them are congruent. The perpendicular bisector of a chord passes through the center of the circle. A radius to a point of tangency is perpendicular to the tangent. Congruent Chords Congruent Chord Theorem Congruent chords intercept congruent arcs on a circle. Congruent arcs on a circle are intercepted by congruent chords. The congruent chord theorem is illustrated in Figure . and .
Congruent chords intercept congruent arcs
This theorem is easily proved by completing the central angle for each arc and then showing the resulting two triangles are congruent. Figure 11.7 shows the congruent chords with the central angles. because they are all radii of the same circle. △APB ≅ △CPD by SSS, and ∠APB ≅ ∠CPB by CPCTC. Since arcs and are intercepted by congruent central angles, the arcs must be congruent. Congruent chords forming congruent central angles in a circle
The congruent chord theorem can be used to help find the measures of arcs and angles. In addition to congruent chords, look out for inscribed angles, central angles, and circle-arc sum relationships around a circle or semicircle. There’s often more than one relationship involved in circle problems. Example 1 In ⊙A, MN = JK. If the measure of is equal to (3x + 8)° and the measure of is equal to (4x − 33)°, what is the measure of each arc?
Solution: The two arcs are intercepted by congruent chords, so their angle measures must be equal. Example 2 ∠ABC is inscribed in ⊙O and measures 35°. If , what is the measure of ?
Solution: The measure of equals twice the measure of the inscribed angle.
Example 3 In ⊙C, chords and are congruent, = 60°, and = 80°. Find m∠ABD.
Solution: ∠ABD is an inscribed angle, so its measure will be .
Parallel Chords Parallel Chords Theorem The two arcs formed between a pair of parallel chords are congruent. If the two arcs formed between a pair of chords are congruent, the chords are parallel. The parallel chord theorem is illustrated in Figure and the arcs between these chords, AC and BD, are congruent. Notice that the location of these two arcs is different from the location of congruent arcs formed by congruent chords.
Congruent arcs between a pair of parallel chords
The parallel chord theorem can be proved by constructing a transversal intersecting the two parallel chords, as shown in Figure 11.9. ∠CDA ≅ ∠BAD because they are alternate interior angles formed by the parallel chords. AC and BD are intercepted by congruent inscribed angles, so the arcs themselves must be congruent.
Parallel chords with transversal
We can apply this theorem to find arc and angle measures in the same way as with the congruent chord theorem. Just do not confuse the two different pairs of congruent arcs. Example 1 In ⊙P, chords and are parallel. If = 118° and = 78°, find .
Solution: and are the congruent arcs between a pair of parallel chords.
If the chords are both parallel and congruent, then there will be two pairs of congruent arcs, one from the congruent chord relationship and one from the parallel chord relationship. Example 2 In circle ⊙P, and . If , find .
Example 3 Trapezoid RSTU has parallel bases and and is inscribed in circle P. If m∠T = 109° and = 92°, find .
Solution: Perpendicular Chords You should be familiar with two perpendicular chord relationships, the chord–perpendicular bisector theorem and the radius-tangent theorem. Chord–Perpendicular Bisector Theorem The perpendicular bisector of any chord passes through the center of the circle. A diameter or radius that is perpendicular to a chord bisects the chord. A diameter or radius that bisects a chord is perpendicular to the chord. You can think of this theorem as linking three properties of a chord: Bisecting another chord Perpendicular to another chord Is a diameter When any two of the above are true, the third is implied to be true.
Figure illustrates the theorem.
is a perpendicular bisector of chord . Therefore passes through center P and is a diameter.
A perpendicular bisector of a chord Example: is a radius of ⊙P and is perpendicular to . If AC = 24 and PC = 13, find PD and DB.
Solution: Radius is perpendicular to , so it must also bisect .
TANGENTS Tangent to a Radius From any point outside a circle, there are exactly two lines that can be drawn tangent to a circle. We call those lines tangents. There will be only one radius of the circle that can be drawn to the point of tangency, and that radius will be perpendicular to the tangent. In Figure 11.11, diameter is perpendicular to tangent at the point of tangency, B. Tangent-Radius Theorem A diameter or radius to a point of tangency is perpendicular to the tangent. A line perpendicular to a tangent at the point of tangency passes through the center of the circle. Tangent perpendicular to a diameter
Whenever you see a radius intersecting a tangent, you should be on the lookout for right triangles, complementary angles, or other applications of right angle relationships. The Pythagorean theorem is often involved. Example 1 Radius and tangent of ⊙D intersect at point G. If DG = 4 and OG = 6, what is the length of OD in simplest radical form?
Solution: The radius and tangent are perpendicular, so △GDO is a right triangle. We can apply the Pythagorean theorem.
The next example is a good exercise in working your way angle by angle toward the desired angle. If the path to the solution is not clear at first, then try to identify any applicable relationship and find any unknown angle. You will usually find that eventually, with a little perseverance, you will get to the goal. Example 2 In ⊙P, is a diameter, is a tangent, , and = 44°. What is the measure of ∠LMG?
Solution: Tangents from a Common Point Tangents from a Common Point Theorem Given a circle and external point P, there are exactly two tangents from P to the circle. Congruent Tangent Theorem Given a circle and external point P, segments between the external point and the point of tangency are congruent. These theorems are illustrated in Figure. Tangents and are both constructed from point Q. The portion of the two tangents between the external point and the point of tangency are congruent, .
Common tangents to circle P
The congruence of the two tangents from a common point can be proved by completing the two triangles shown in Figure. Radii , , and ∠KLO and ∠KMO are right angles because they are formed by a radius and tangent. △KMO ≅ △KLO by the HL postulate, so by CPCTC.
Congruent tangents
and
A triangle circumscribed about a circle is comprised of a series of tangents from common points—the vertices of the triangles. We can use the congruent tangents to help find the perimeter of the triangle. Example 1 △RST is circumscribed about a circle. If SX = 6, XR = 7, and TZ = 8, what is the perimeter of △RST? Solution: SY = SX, TY = TZ, and ZR = XY because each pair are tangents from the same point. The perimeter is therefore 2 · 6 + 2 · 7 + 2 · 8 = 42. The next example combines several of the circle theorems we have seen so far. Example 2 Tangents and are drawn from point T to circle A. is a diameter of circle A. If m∠T = 46°, what is the measure of ?
Solution: In the next section, we will see a set of angle theorems involving tangents, secants, and chords that could also have been applied to this example.
ANGLE-ARC RELATIONSHIPS WITH CHORDS, TANGENTS, AND SECANTS Key Ideas Angles formed by intersecting chords, secants, and tangents have the following relationships with the intercepted arcs:
Angles with Vertex Inside the Circle When two chords intersect inside a circle, they form a pair of congruent vertical angles that intercept two arcs. Angle-Arc Theorem for Intersecting Chords (Vertex Inside the Circle) The measures of a pair of vertical angles formed by two intersecting chords are equal to one-half the sum of the measures of the intercepted arcs. Figure shows chords and intersecting inside a circle. Vertical angles ∠1 and ∠2 intercept arcs and .
Vertical angles ∠2 and ∠3 intercept arcs and .
The angle measures are calculated from the arcs as follows: Angles formed by intersecting chords
This theorem can be proved by sketching to complete triangle △ADE within the circle as shown in Figure. ∠1 is an exterior angle of △ADE. By the exterior angle theorem, m∠1 = m∠4 + m∠5. ∠5 is an inscribed angle, equal in measure to . ∠4 is an inscribed angle equal in measure to . Combining these equations gives .
The same procedure can be used to show the matching relationship with other pairs of vertical angles. Proving the angle-arc theorem for intersecting chords Example 1 Chords and intersect at G in circle P. = 41° and = 133°. Find m∠AGC, m∠BGD, m∠AGD, and m∠BGC.
Solution: Example 2 In ⊙A, is a diameter, = 105°, and m∠GKM = 64°. Find and .
We now have three out of four arcs around the circle and can solve for the fourth. Confusing central angles with angles formed by two chords is a common error. Be careful not to assume the angle formed by the chords is equal to the measure of the intercepted arc—that is the relationship for central angles only. Math Fact The theorem that states that the measure of a central angle equals the measure of the intercepted arc is just a special case of the intersecting chords theorem. When the chords are diameters, the vertical angles formed are central angles. Angles with Vertex on the Circle Angle-Arc Theorem for Chords and Tangents (Vertex on the Circle) The measure of the angle formed by two chords, or a chord and tangent, with the vertex on the circle is equal to the measure of the intercepted arc. Figure illustrates the three cases of the theorem. Figure 11.16a shows ∠ABC formed by tangent and chord . The measure of ∠B equals the measure of arc .
Figure b shows the special case where the chord is a diameter. The angle formed is equal to (180°) or 90°. When the angle is formed by two chords, shown in Figure 11.16c, we get the familiar theorem that the measure of an inscribed angle is equal to the measure of the intercepted arc.
Angle with vertex on the circle Example 1 In circle P, chord intersects tangent at point T. = 152°. What is the measure of ∠RTS ?
Solution: Example 2 In circle P, tangent intersects chord at T. If = 116°, what is the measure of ∠RTV?
Solution: The intercepted arc is , so we begin by finding the measure of .
Angles with Vertex Outside the Circle When an angle is formed by two tangents, two secants, or a tangent and secant, the vertex lies outside the circle and we have the following theorem: Angle-Arc Theorem for Tangents and Secants (Vertex Outside) The measure of the angle formed by two secants, two tangents, or a secant and tangent is equal to the difference of the measures of the two intercepted arcs. Figure shows the angle-arc relationship for the three cases where the vertex is outside the circle. Angle-arc relationships, (a) two tangents, (b) a tangent and a secant, and (c) two secants Example 1 In ⊙P, secants and intersect at R. If the measure of = 140° and the measure of arc is 42°, what is the measure of ∠R? If m∠R = 36° and = 48°, find .
Solution: The angle measure equals the difference between the measures of the intercepted arcs. Here we work backward from the relationship to find the angle.
When the angle is formed by two tangents, only one of the arc measures is needed to find the angle. We can find the other arc measure using the fact that the two measures must sum to 360°. Example 2 and are tangent to ⊙P at points M and N. If the measure of = 115°, what is the measure of ∠A?
If the measure of ∠A is 32°, what are the measures of arcs and ? Solution: First we need to find . Now apply the tangent-arc relationship. Here both arc measures are unknown. However, they are related by the fact that their sum equals 360°. So we really have only one variable. Let = x° and . It doesn’t matter which arc we define as x. We will still come up with the same answer in the end. Either way, watch out for the signs—there will be multiple negative signs in the equation. Now substitute to find each arc. The next example brings together several of the circle theorems we have seen. Example 4 In circle P, m∠R = 47° and = 86°. Are points T, P, and U collinear? Is ST = SU? Justify your reasoning. Solution: The strategy here is to find all the arc measures. For T, P, and U to be collinear, they must form a straight line through the center and lie on a diameter. The measure of arc would have to be 180°. If ST = SU, the two chords would have to intercept congruent arcs and . Using the tangent-secant relationship, we can find .
So points T, P, and U must be collinear.
We can now find the measure of by setting the sum of the arc measures equal to 360°.
Since arcs and are not congruent, we know that and are not congruent and ST ≠ SU.
SEGMENT RELATIONSHIPS IN INTERSECTING CHORDS, TANGENTS, AND SECANTS Key Ideas The relationships between segment lengths formed by intersecting chords, tangents, and secants are shown in the following table:
Segments Formed by Intersecting Chords Theorem When two chords intersect in a circle, the products of the parts of the two chords are equal. Figure shows the two segments formed within each of two intersecting chords.
The product of the parts are equal, giving the relationship a · b = c · d. Product of the parts of intersecting chords are equal
The theorem is based on the similar triangles formed by completing the two triangles, which is illustrated in Figure 11.19. ∠C ≅ ∠A because they intercept the same arc. ∠B ≅ ∠D for the same reason. △ADE ≅ △CBE by AA.
Corresponding parts are in the same ratio, which gives:
Proof of the intersecting chord theorem Example 1 Chords and intersect at E in ⊙O. If BE = 6, AE = 12, and DE = 8, find EC. If DC = 12, BE = 4, and AE = 9, find DE and EC. Solution: Apply the product of the parts relationship. We are given the entire length of DE instead of the length of the parts. However, we can use the fact that the sum of the parts equals the whole. Example 2 In ⊙O, chords and intersect at E. If AE = 4, AB = 12, BE = 15, and CD = 8, what are the lengths of and ?
Solution: △ABE ~ △CDE by AA because ∠A and ∠C both intercept and ∠D and ∠B both intercept .
Example 3 is a diameter of ⊙P and bisects . If GD = 12 and AD = 15, find BG and BC.
Solution: We know △AGD and △CGB are similar triangles. Vertical angles ∠AGD and ∠BGC are congruent. ∠C ≅ ∠D because they are inscribed angles intercepting the same arc. So △AGD ~ △BGC. The diameter bisects , so it must be perpendicular to . We can use the Pythagorean theorem to find AG:
Since is bisected at G,
We now have enough information to apply a similarity relationship.
Segments Formed by Tangents and Secants Theorem When two secants from a common point external to a circle intersect that circle, the products of the outer segments and the entire secants are equal. When a secant and tangent from a common point external to a circle intersect that circle, the products of the outer segments and the entire secant or tangent are equal. Segment relationship between secants and tangent intersecting a circle
When a secant intersects a circle, it is divided into a segment outside the circle and a segment within the circle. Figure 11.20a shows two segments intersecting circle O from point P.
The outer part of secant is , and the outer part of secant is .
The secant-segment theorem states:
The relationship is similar when a tangent and secant meet at a common point outside the circle. The only difference is that the entire length is outside the circle as shown in Figure 11.20b. So the outer length and whole length are the same, giving the relationship The theorem also holds for two tangents as shown in Figure c.
The relationship is Example 1 Secants and intersect ⊙P at B and D. If AB = 9, BC = 3, and AD = 6, find DE. If BC = 9, AD = 12, and AE = 30, find AB.
Solution: Let AB = x and AC = x + BC Example 2 Secant intersects ⊙P at N and G. Tangent intersects ⊙P at I. If KN = 5 and NG = 7, find IK. Express your answer in simplest radical form. Solution:
AREA, CIRCUMFERENCE, AND ARC LENGTH Key Ideas The radian is a unit of angle measure. 2π radians is equal to one complete revolution around a circle, or 360°. You can convert between units using the following formulas:
radians = · degrees degrees = · radians The length of an arc and the area of a sector formed by that arc and its central angle can be calculated from the measure of the central angle. Multiply the circumference or area of the circle by the fraction of the circle spanned by the central angle.
θ is the central angle measure, r is the radius of the circle, c is the circumference of the circle, and A is the area of the circle. Area and Circumference The circumference, C, of a circle is the length around the circle. You can think of it as the length of a piece of string that would wrap exactly once around the circle. The area, A, of a circle is the area enclosed by the circle or the area of the interior region. Formulas In a circle of radius r and diameter d, Circumference of a circle: C = 2πr = πd
Area of a circle: When working with area and circumference, be sure to read the question carefully to determine if answers should be rounded or be in terms of π. Example 1 Find the area and circumference of a circle whose diameter is 12 in. Express your answer in terms of π. Solution: The radius of the circle equals the diameter, or 6 in. We can also work backward to find the radius or diameter from the circumference or area. Math Fact When working with the quantity π, you may be asked to express your answer either in terms of π or as a rounded value. When giving a rounded value, you should always use the π button on your calculator instead of 3.14. This way you will be certain to retain enough precision in your answer. Example 2 Anna has a round table for which she would like to buy a tablecloth. She wants to get an accurate measurement of the table’s diameter. So she wraps a tape measure around the table and measures the length around the table to be inches. What is the diameter of the table in feet? Round to the nearest foot. Solution: We are given a circumference of inches.
So we can use the circumference formula to solve for the diameter.
Multiply by to find the diameter in feet.
Example 3 An athletic field is in the shape of a square with semicircular ends. If the distance between the two semicircles is 30 meters, what is the area of the athletic field in terms of π? Solution: All sides of a square are congruent, so the sides of the square that form the diameter of the semicircles are also 30 meters. The radius of the two semicircles is 15 meters.
The area of each semicircle will be half of the area of a full circle. Arc Length and Radian Measure Arcs have length, which equals the length of a string wrapped around a circle starting and ending at the two endpoints of the arc. We will refer to the arc length with the variable s.
Angle measures are often represented with the Greek letter θ (theta).
Figure shows , angle measure θ, and arc length s.
Arc length s versus arc angle measure θ
The circumference is the length of an arc spanning the entire circle. If the arc spans less than the full circle, the arc length is reduced proportionally. The fraction of the circle spanned by a central angle measuring θ degrees is θ/360°. This fact makes the intercepted arc length equal to θ/360° · circumference.
Degrees are not the only unit of measure for angle. The other unit of angle measure you need to know is the radian. Definition Radian—A unit of angle measure. In a circle, a central angle measuring 1 radian intercepts an arc with length equal to the radius of the circle. Figure 11.22 illustrates the concept of the radian. ∠APB is a central angle whose measure is 1 radian and intercepts . The length of equals s, which equals r in this case. 2π radians is equivalent to 1 full revolution. An arc intercepted by a central angle measuring 1 radian
The following formulas are used to convert between radians and degrees. These are found on the Common Core reference sheet.
Example 1 The radius of a circle is 4 inches. Find the length of the arc intercepted by a central angle measuring 1.2 radians. Solution: Example 2 The radius of a circle is 3 cm. What is the length of the arc intercepted by a central angle measuring 42°? Round your answer to the nearest tenth of a centimeter. Solution: The angle is given in degrees, so use the formula Example 3 ∠FKG is an inscribed angle measuring radians. What is the length of if the radius of the circle is 10 meters? Express your answer in terms of π. Solution: Sketching the figure will help here: The central angle that intercepts the same arc will have a measure twice that of the intercepted angle, or radians. Example 4 Frannie is planting a flower bed in the area around a corner of her backyard fence. The two sections of fence come together at an angle of radians. She wants the flower bed to have a radius of 6 feet. Frannie wants to put a brick border around the flower bed. What will be the length of the border to the nearest inch? Solution: The region is bounded by a circular arc with the angle given in radians. So we use the formula
We convert to inches using the ratio :
The border will have a length of 302 in. Sectors A sector is the region bounded by a circle and a central angle. Figure 11.23 shows sector APB, the shaded region, in circle P. Sector APB The area of a sector is equal to the fraction of the circle spanned by the central angle multiplied by the area of the circle. Formula for Area of a Sector where θ is the measure of the central angle in degrees where θ is the measure of the central angle in radians Example 1 Find the area of a sector of a circle if the radius is 12 cm and the sector is bounded by a central angle measuring 0.25 radians. Solution: Example 2 In a circle of radius 8 in., a sector has an area of 40 in.2 What is the measure of the central angle that bounds the sector? Express your answer in radians. Solution: Example 3 A spinner for a board game has a radius of 9 inches and is divided into 5 regions. The central angles that bound regions 2 through 5 each measure 60°. What is the area of region 1? Round your answer to the nearest 0.1 square inch. Solution: The central angle that bounds region 1 is equal to 360° minus the sum of the remaining central angles. Example 4 The discus-throwing area at a track and field match is in the shape of a sector of a circle. The length along each side is 80 meters, and the angle is 34.9°. What is the area of the sector? Round your answer to the nearest tenth of a square meter. Officials want to mark off the entire area with a chalk-line boundary. What is the total length of the chalk-line? Round your answer to the nearest tenth of a meter.
Solution: The region is a circular sector with the angle given in degrees, so we can use the formula The curved part of the sector is an arc, whose length is given by Now add the two straight 80-meter segments. Total length = 48.7 + 80 + 80 = 208.7 m. Arc Length and Sector Area in Similar Circles All circles are similar and differ only in the length of their radii. If two circles have radii in a ratio of r1/r2 = k, we can determine the ratio of arc lengths and sector areas given the same central angle.
The ratio of arc lengths is
Since the central angle measures are congruent, .
The ratio is the constant of proportionality between the two circles, or the dilation factor k.
By substituting we get Arc Length Theorem for Similar Circles The ratio of arc lengths, given congruent central angles, is equal to the ratio of the radii of the circles. The result is the same whether the central angles are measured in radians or degrees. By applying the same process to sector areas we find Sector Area Theorem for Similar Circles The ratio of sector areas, given congruent central angles, is equal to the ratio of the radii squared. As with arc length, the result is the same whether central angles are measured in radians or degrees.
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