10.1 Tangents to Circles

Some definitions you need Circle – set of all points in a plane that are equidistant from a given point called a center of the circle. A circle with center P is called “circle P”, or P. The distance from the center to a point on the circle is called the radius of the circle. Two circles are congruent if they have the same radius.

Some definitions you need The distance across the circle, through its center is the diameter of the circle. The diameter is twice the radius. The terms radius and diameter describe segments as well as measures.

Some definitions you need A radius is a segment whose endpoints are the center of the circle and a point on the circle. QP, QR, and QS are radii of Q. All radii of a circle are congruent.

Some definitions you need A chord is a segment whose endpoints are points on the circle. PS and PR are chords. A diameter is a chord that passes through the center of the circle. PR is a diameter.

Some definitions you need A secant is a line that intersects a circle in two points. Line k is a secant. A tangent is a line in the plane of a circle that intersects the circle in exactly one point. Line j is a tangent.

Ex. 1: Identifying Special Segments and Lines Tell whether the line or segment is best described as a chord, a secant, a tangent, a diameter, or a radius of C. a.AD b.CD c.EG d.HB

Ex. 1a: Identifying Special Segments and Lines Tell whether the line or segment is best described as a chord, a secant, a tangent, a diameter, or a radius of C. a.AD – Diameter because it contains the center C. b.CD c.EG d.HB

Ex. 1b: Identifying Special Segments and Lines Tell whether the line or segment is best described as a chord, a secant, a tangent, a diameter, or a radius of C. a.AD – Diameter because it contains the center C. b.CD– radius because C is the center and D is a point on the circle.

Ex. 1: Identifying Special Segments and Lines Tell whether the line or segment is best described as a chord, a secant, a tangent, a diameter, or a radius of C. c. EG – a tangent because it intersects the circle in one point.

Ex. 1c: Identifying Special Segments and Lines Tell whether the line or segment is best described as a chord, a secant, a tangent, a diameter, or a radius of C. c.EG – a tangent because it intersects the circle in one point. d.HB is a chord because its endpoints are on the circle.

More information you need-- In a plane, two circles can intersect in two points, one point, or no points. Coplanar circles that intersect in one point are called tangent circles. Coplanar circles that have a common center are called concentric. 2 points of intersection.

Tangent circles A line or segment that is tangent to two coplanar circles is called a common tangent. A common internal tangent intersects the segment that joins the centers of the two circles. A common external tangent does not intersect the segment that joins the center of the two circles. Internally tangent Externally tangent

Concentric circles Circles that have a common center are called concentric circles. Concentric circles No points of intersection

Ex. 2: Identifying common tangents Tell whether the common tangents are internal or external.

Ex. 2: Identifying common tangents Tell whether the common tangents are internal or external. The lines j and k intersect CD, so they are common internal tangents.

Ex. 2: Identifying common tangents Tell whether the common tangents are internal or external. The lines m and n do not intersect AB, so they are common external tangents. In a plane, the interior of a circle consists of the points that are inside the circle. The exterior of a circle consists of the points that are outside the circle.

Using properties of tangents The point at which a tangent line intersects the circle to which it is tangent is called the point of tangency.

Theorem 10.1 If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. If l is tangent to Q at point P, then l ⊥ QP. l

Theorem 10.2 In a plane, if a line is perpendicular to a radius of a circle at its endpoint on a circle, then the line is tangent to the circle. If l ⊥ QP at P, then l is tangent to Q. l

Ex. 3: Verifying a Tangent to a Circle You can use the Converse of the Pythagorean Theorem to tell whether EF is tangent to D. Because 11 2 _ 60 2 = 61 2, ∆DEF is a right triangle and DE is perpendicular to EF. So by Theorem 10.2; EF is tangent to D.

Ex. 4: Finding the radius of a circle You are standing at C, 8 feet away from a grain silo. The distance from you to a point of tangency is 16 feet. What is the radius of the silo? First draw it. Tangent BC is perpendicular to radius AB at B, so ∆ABC is a right triangle; so you can use the Pythagorean theorem to solve.

Solution: (r + 8) 2 = r Pythagorean Thm. Substitute values c 2 = a 2 + b 2 r r + 64 = r Square of binomial 16r + 64 = r = 192 r = 12 Subtract r 2 from each side. Subtract 64 from each side. Divide. The radius of the silo is 12 feet.

Theorem 10.3 If two segments from the same exterior point are tangent to the circle, then they are congruent. IF SR and ST are tangent to P, then SR  ST.

Proof of Theorem 10.3 Given: SR is tangent to P at R. Given: ST is tangent to P at T. Prove: SR  ST

Ex. 5: Using properties of tangents AB is tangent to C at B. AD is tangent to C at D. Find the value of x. x 2 + 2

Solution: x = x Two tangent segments from the same point are  Substitute values AB = AD 9 = x 2 Subtract 2 from each side. 3 = xFind the square root of 9. The value of x is 3 or -3.

10.2 Arcs and Chords

Using Arcs of Circles In a plane, an angle whose vertex is the center of a circle is a central angle of the circle. If the measure of a central angle,  APB is less than 180 °, then A and B and the points of P

Using Arcs of Circles in the interior of  APB form a minor arc of the circle. The points A and B and the points of P in the exterior of  APB form a major arc of the circle. If the endpoints of an arc are the endpoints of a diameter, then the arc is a semicircle.

Naming Arcs Arcs are named by their endpoints. For example, the minor arc associated with  APB above is. Major arcs and semicircles are named by their endpoints and by a point on the arc. 60 ° 180 °

Naming Arcs For example, the major arc associated with  APB is. here on the right is a semicircle. The measure of a minor arc is defined to be the measure of its central angle. 60 ° 180 °

Naming Arcs For instance, m = m  GHF = 60 °. m is read “the measure of arc GF.” You can write the measure of an arc next to the arc. The measure of a semicircle is always 180°. 60 ° 180 °

Naming Arcs The measure of a major arc is defined as the difference between 360° and the measure of its associated minor arc. For example, m = 360° - 60° = 300°. The measure of the whole circle is 360°. 60 ° 180 °

Ex. 1: Finding Measures of Arcs Find the measure of each arc of R. a. b. c. 80 °

Ex. 1: Finding Measures of Arcs Find the measure of each arc of R. a. b. c. Solution: is a minor arc, so m = m  MRN = 80 ° 80 °

Ex. 1: Finding Measures of Arcs Find the measure of each arc of R. a. b. c. Solution: is a major arc, so m = 360 ° – 80 ° = 280 ° 80 °

Ex. 1: Finding Measures of Arcs Find the measure of each arc of R. a. b. c. Solution: is a semicircle, so m = 180 ° 80 °

Note: Two arcs of the same circle are adjacent if they intersect at exactly one point. You can add the measures of adjacent areas. Postulate 26—Arc Addition Postulate. The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. m = m + m

Ex. 2: Finding Measures of Arcs Find the measure of each arc. a. b. c. m = m + m = 40 ° + 80° = 120° 40 ° 80 ° 110 °

Ex. 2: Finding Measures of Arcs Find the measure of each arc. a. b. c. m = m + m = 120 ° + 110° = 230° 40 ° 80 ° 110 °

Ex. 2: Finding Measures of Arcs Find the measure of each arc. a. b. c. m = 360 ° - m = 360 ° - 230° = 130° 40 ° 80 ° 110 °

Ex. 3: Identifying Congruent Arcs Find the measures of the blue arcs. Are the arcs congruent? and are in the same circle and m = m = 45 °. So,  45 °

Ex. 3: Identifying Congruent Arcs Find the measures of the blue arcs. Are the arcs congruent? and are in congruent circles and m = m = 80 °. So,  80 °

m = m = 65°, but and are not arcs of the same circle or of congruent circles, so and are NOT congruent. Ex. 3: Identifying Congruent Arcs Find the measures of the blue arcs. Are the arcs congruent? 65 °

Using Chords of Circles A point Y is called the midpoint of if . Any line, segment, or ray that contains Y bisects.

Theorem 10.4 In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.  if and only if 

Theorem 10.5 If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. , 

Theorem 10.5 If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. is a diameter of the circle.

Ex. 4: Using Theorem 10.4 You can use Theorem 10.4 to find m. Because AD  DC, and . So, m = m 2x = x + 40Substitute x = 40 Subtract x from each side. 2x ° (x + 40) °

In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center. AB  CD if and only if EF  EG. Theorem 10.7

Ex. 7: Using Theorem 10.7 AB = 8; DE = 8, and CD = 5. Find CF.

Ex. 7: Using Theorem 10.7 Because AB and DE are congruent chords, they are equidistant from the center. So CF  CG. To find CG, first find DG. CG  DE, so CG bisects DE. Because DE = 8, DG = =4.

Ex. 7: Using Theorem 10.7 Then use DG to find CG. DG = 4 and CD = 5, so ∆CGD is a right triangle. So CG = 3. Finally, use CG to find CF. Because CF  CG, CF = CG = 3

10.3 Inscribed Angles

Using Inscribed Angles An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. The arc that lies in the interior of an inscribed angle and has endpoints on the angle is called the intercepted arc of the angle.

Theorem 10.8: Measure of an Inscribed Angle If an angle is inscribed in a circle, then its measure is one half the measure of its intercepted arc. m  ADB = ½m

Ex. 1: Finding Measures of Arcs and Inscribed Angles Find the measure of the blue arc or angle. m = 2m  QRS = 2(90°) = 180°

Ex. 1: Finding Measures of Arcs and Inscribed Angles Find the measure of the blue arc or angle. m = 2m  ZYX = 2(115°) = 230°

Ex. 1: Finding Measures of Arcs and Inscribed Angles Find the measure of the blue arc or angle. m = ½ m ½ (100°) = 50° 100°

Ex. 2: Comparing Measures of Inscribed Angles Find m  ACB, m  ADB, and m  AEB. The measure of each angle is half the measure of m = 60°, so the measure of each angle is 30°

Theorem 10.9 If two inscribed angles of a circle intercept the same arc, then the angles are congruent.  C   D

Ex. 3: Finding the Measure of an Angle It is given that m  E = 75 °. What is m  F?  E and  F both intercept, so  E   F. So, m  F = m  E = 75° 75°

Ex. 5: Using Theorems and Find the value of each variable. AB is a diameter. So,  C is a right angle and m  C = 90 ° 2x° = 90° x = 45 2x°

Ex. 5: Using Theorems and Find the value of each variable. DEFG is inscribed in a circle, so opposite angles are supplementary. m  D + m  F = 180° z + 80 = 180 z = ° 80° y°y° z°z°

Ex. 5: Using Theorems and Find the value of each variable. DEFG is inscribed in a circle, so opposite angles are supplementary. m  E + m  G = 180° y = 180 y = ° 80° y°y° z°z°

Ex. 6: Using an Inscribed Quadrilateral In the diagram, ABCD is inscribed in circle P. Find the measure of each angle. ABCD is inscribed in a circle, so opposite angles are supplementary. 3x + 3y = 180 5x + 2y = 180 3y° 2y° To solve this system of linear equations, you can solve the first equation for y to get y = 60 – x. Substitute this expression into the second equation. 3x° 2x°

Ex. 6: Using an Inscribed Quadrilateral 5x + 2y = x + 2 (60 – x) = 180 5x – 2x = 180 3x = 60 x = 20 y = 60 – 20 = 40 Write the second equation. Substitute 60 – x for y. Distributive Property. Subtract 120 from both sides. Divide each side by 3. Substitute and solve for y.  x = 20 and y = 40, so m  A = 80°, m  B = 60°, m  C = 100°, and m  D = 120°

10.4 Other Angle Relationships in Circles

Using Tangents and Chords You know that measure of an angle inscribed in a circle is half the measure of its intercepted arc. This is true even if one side of the angle is tangent to the circle. m  ADB = ½m

Theorem If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc. m  1= ½m m  2= ½m

Ex. 1: Finding Angle and Arc Measures Line m is tangent to the circle. Find the measure of the red angle or arc. Solution: m  1= ½ m  1= ½ (150 °) m  1= 75 ° 150°

Ex. 1: Finding Angle and Arc Measures Line m is tangent to the circle. Find the measure of the red angle or arc. Solution: m = 2(130 °) m = 260 ° 130°

Ex. 2: Finding an Angle Measure In the diagram below, is tangent to the circle. Find m  CBD Solution: m  CBD = ½ m 5x = ½(9x + 20) 10x = 9x +20 x = 20  m  CBD = 5(20 °) = 100° (9x + 20)° 5x° D

Lines Intersecting Inside or Outside a Circle If two lines intersect a circle, there are three (3) places where the lines can intersect. on the circle

Inside the circle

Outside the circle

Lines Intersecting You know how to find angle and arc measures when lines intersect ON THE CIRCLE. You can use the following theorems to find the measures when the lines intersect INSIDE or OUTSIDE the circle.

Theorem If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle. m  1 = ½ m + mm  2 = ½ m + m

Theorem If a tangent and a secant, two tangents or two secants intercept in the EXTERIOR of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs. m  1 = ½ m( - m )

Theorem If a tangent and a secant, two tangents or two secants intercept in the EXTERIOR of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs. m  2 = ½ m( - m )

Theorem If a tangent and a secant, two tangents or two secants intercept in the EXTERIOR of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs. m  3 = ½ m( - m ) 3

Ex. 3: Finding the Measure of an Angle Formed by Two Chords Find the value of x Solution: x ° = ½ (m +m x ° = ½ (106° + 174°) x = 140 Apply Theorem Substitute values Simplify 174 ° 106 ° x°x°

Ex. 4: Using Theorem Find the value of x Solution: 72 ° = ½ (200° - x°) 144 = x ° - 56 = -x 56 = x Substitute values. Subtract 200 from both sides. Multiply each side by 2. m  GHF = ½ m( - m ) Apply Theorem Divide by -1 to eliminate negatives. 200 ° x°x° 72 °

Ex. 4: Using Theorem Find the value of x Solution: = ½ ( ) = ½ (176) = 88 Substitute values. Multiply Subtract m  GHF = ½ m( - m ) Apply Theorem x°x° 92 ° Because and make a whole circle, m =360 °-92°=268°

10.5 Segment Lengths in Circles

Theorem If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. EA EB = EC ED

Ex. 1: Finding Segment Lengths Chords ST and PQ intersect inside the circle. Find the value of x. RQ RP = RS RTUse Theorem Substitute values.9 x = 3 6 9x = 18 x = 2 Simplify. Divide each side by 9.

Using Segments of Tangents and Secants In the figure shown, PS is called a tangent segment because it is tangent to the circle at an end point. Similarly, PR is a secant segment and PQ is the external segment of PR.

Theorem If two secant segments share the same endpoint outside a circle, then the product of the length of one secant segment and the length of its external segment equals the product of the length of the other secant segment and the length of its external segment. EA EB = EC ED

Theorem If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the length of the secant segment and the length of its external segment equal the square of the length of the tangent segment. (EA) 2 = EC ED

Ex. 2: Finding Segment Lengths Find the value of x. RP RQ = RS RTUse Theorem Substitute values.9(11 + 9)=10(x + 10) 180 = 10x = 10x Simplify. Subtract 100 from each side. 8 = x Divide each side by 10.

10.6 Equations of Circles

Finding Equations of Circles You can write an equation of a circle in a coordinate plane if you know its radius and the coordinates of its center.

Finding Equations of Circles Suppose the radius is r and the center is (h, k). Let (x, y) be any point on the circle. The distance between (x, y) and (h, k) is r, so you can use the Distance Formula. (Told you it wasn’t going away).

Finding Equations of Circles Square both sides to find the standard equation of a circle with radius r and center (h, k). (x – h) 2 + (y – k) 2 = r 2 If the center is at the origin, then the standard equation is x 2 + y 2 = r 2.

Ex. 1: Writing a Standard Equation of a Circle Write the standard equation of the circle with a center at (-4, 0) and radius 7. (x – h) 2 + (y – k) 2 = r 2 Standard equation of a circle. [(x – (-4)] 2 + (y – 0) 2 = 7 2 Substitute values. (x + 4) 2 + (y – 0) 2 = 49 Simplify.

Ex. 2: Writing a Standard Equation of a Circle The point (1, 2) is on a circle whose center is (5, -1). Write a standard equation of the circle. r = r = 5 Use the Distance Formula Substitute values. Simplify. Addition Property Square root the result.

Ex. 2: Writing a Standard Equation of a Circle The point (1, 2) is on a circle whose center is (5, -1). Write a standard equation of the circle. (x – h) 2 + (y – k) 2 = r 2 Standard equation of a circle. [(x – 5)] 2 + [y –(-1)] 2 = 5 2 Substitute values. (x - 5) 2 + (y + 1) 2 = 25 Simplify.

Graphing Circles If you know the equation of a circle, you can graph the circle by identifying its center and radius.

Ex. 3: Graphing a circle The equation of a circle is (x+2) 2 + (y-3) 2 = 9. Graph the circle. First rewrite the equation to find the center and its radius. (x+2) 2 + (y-3) 2 = 9 [x – (-2)] 2 + (y – 3) 2 =3 2 The center is (-2, 3) and the radius is 3.

To graph the circle, place the point of a compass at (-2, 3), set the radius at 3 units, and swing the compass to draw a full circle.

Ex. 4: Applying Graphs of Circles 1.Rewrite the equation to find the center and radius. –(x – h) 2 + (y – k) 2 = r 2 –(x - 13) 2 + (y - 4) 2 = 16 –(x – 13) 2 + (y – 4) 2 = 4 2 –The center is at (13, 4) and the radius is 4. The circle is shown on the next slide.