Circles Vocabulary Interior Exterior Radius Diameter Chord Secant

Circles

Vocabulary • • Interior Exterior Radius Diameter Chord Secant Tangent

Lesson Quiz: Part I 1. Identify each line or segment that intersects � Q. chords VT and WR secant: VT tangent: s diam. : WR radii: QW and QR

Circles • Congruent • Concentric • Tangent – Internally tangent – Externally tangent

Tangent Line Radius

Tangents • Tangent – Perpendicular to radius • Example: Visible distance to horizon from Mt Everest – Mt Everest is 29, 000 ft above sea level • Find: EH – 2 tangents from external point • Same measure

12 -2 Arcs & Chords • Types of Arcs – Minor Arc • < 180 o – Major Arc • >180 o – Semi-Circle • = 180 o • Central Angle: a angle whose vertex is at the center of a circle

a. m FMC = 0. 30(360 ) = 108 Central is 30% of the �. c. m EMD = 0. 10(360 ) b. m. AHB = 360° – m AMB m AHB = 360° – 0. 25(360 ) = 36 = 270 Central is 10% of the �.

Solving for Arcs: • Adjacent Arcs: • Arc Addition Postulate:

Find each measure. m. LJN m. JKL m KPL = 180° – (40 + 25)° m. KL = 115° m. JKL = m. JK + m. KL = 25° + 115° = 140° Arc Add. Post. Substitute. Simplify.

• Congruent Arcs – 2 Arcs with the same measure – Central Angles – Chords

Congruent Arcs • Examples • Find RT • Find m. CD

Radii & Chords • If radius is perpendicular to chord – Bisects Chord & Arc • A perpendicular bisector of chord is a radius

Using Radii and Chords Find NP. Step 1 Draw radius RN. RN = 17 Radii of a �are . Step 2 Use the Pythagorean Theorem. SN 2 + RS 2 = RN 2 SN 2 + 82 = 172 SN 2 = 225 SN = 15 Substitute 8 for RS and 17 for RN. Subtract 82 from both sides. Take the square root of both sides. Step 3 Find NP. NP = 2(15) = 30 RM NP , so RM bisects NP.

Find QR to the nearest tenth. Step 1 Draw radius PQ. PQ = 20 Radii of a �are . Step 2 Use the Pythagorean Theorem. TQ 2 + PT 2 = PQ 2 Substitute 10 for PT and 20 for PQ. TQ 2 + 102 = 202 Subtract 102 from both sides. TQ 2 = 300 Take the square root of both sides. TQ 17. 3 Step 3 Find QR. QR = 2(17. 3) = 34. 6 PS QR , so PS bisects QR.

Examples

Inscribed Angles • Inscribed angle – Vertex on circle – Sides contain chords • Measure of inscribed angle = ½ measure of arc – m<E = ½(m. DF)

Inscribed Angles • If inscribed angle arcs are congruent – Intercept same arc or congruent arcs – THEN: Inscribed angles are congruent

Example • Find m. BC • Find m<ECD • Find m<DEC

Example 2: Hobby Application An art student turns in an abstract design for his art project. Find m DFA = m DCF + m CDF = 115°

Inscribed Angle • Inscribed angle subtends a semicircle if and only if the angle is a right angle • Example:

Quadrilaterals • Opposite Angles Add to 180

Example: Finding Angle Measures in Inscribed Quadrilaterals Find the angle measures of GHJK. Step 1 Find the value of b. . m G + m J = 180 GHJK is inscribed in a � 3 b + 25 + 6 b + 20 = 180 Substitute the given values. 9 b + 45 = 180 Simplify. 9 b = 135 Subtract 45 from both sides. b = 15 Divide both sides by 9.

Angle Relationships • Tangent and a secant/chord – Measure of angle is ½ intercepted arc measure – Measure of the arc is twice the measure of angle • Example: – Find m<BCD – Find measure of arc AB

Interior Angle • Intersect inside circle – Measure of vertical angles is ½ sum of arcs • Example: – Find m<PQT

Exterior Angle •

Examples • Find x 2. 1. 3.

4. Find m. CE. 12°

Arc Length/Sector/Segment

Sectors & Arc Length Geometry 12. 3 • Sector of a circle – 2 radii & arc – The pie shaped slice of the circle • Area of sector is percent area of circle based on arc or central angle: • A= Area, r= radius, m= measure of arc/angle

Segments of Circles • Segment of circle – Area of arc bounded by chord • Finding Area of Segment

Segment Examples Segment: – segment RST - segment DEF Area sector-Area of Triangle

Arc Length • Distance along the arc (circumference) – Measured in linear units – L= length, r= radius, m= measure of arc/angle • Example: – Measure of GH

Segment Relationships Geometry 12. 6 J Example: J

Example: Art Application The art department is contracted to construct a wooden moon for a play. One of the artists creates a sketch of what it needs to look like by drawing a chord and its perpendicular bisector. Find the diameter of the circle used to draw the outer edge of the moon.

Example 2 Continued 8 (d – 8) = 9 9 8 d – 64 = 81 8 d = 145

External Secant Relationships Example:

External Tan/Sec Relationship Example:

Lesson Review: Part I 1. Find the value of d and the length of each chord. d=9 ZV = 17 WY = 18 2. Find the diameter of the plate.

Lesson Review: Part II 3. Find the value of x and the length of each secant segment. x = 10 QP = 8 QR = 12 4. Find the value of a. 8

Equation for Circle Geometry 12. 7 & Algebra 2 12. 2 • (x – h)2 + (y – k)2 = r 2 – h is the x coordinate of the center point – k is the y coordinate of the center point – r is the radius • To determine if point is inside/on/outside – Plug x and y of point into circle equation • h & k are the CENTER POINT coordinates – Compare result to r 2 • If < pt is inside : if > pt is outside : if = pt is on

Distance Example: Consumer Application Use the map and information given in Example 3 on page 730. Which homes are within 4 miles of a restaurant located at (– 1, 1)? The circle has a center (– 1, 1) and radius 4. The points insides the circle will satisfy the 2 2 2 inequality (x + 1) + (y – 1) < 4. Points B, C, D and E are within a 4 -mile radius. Check Point F(– 2, – 3) is near the boundary. (– 2 + 1)2 + (– 3 – 1)2 < 42 2 2 (– 1) + (– 4) < 4 1 + 16 < 16 2 x Point F (– 2, – 3) is not inside the circle.

Finding center & radius • Given 2 endpoints – Find center point • X coordinate is (x 1+x 2)/2 • Y coordinate is (y 1+y 2)/2 – Use center point coordinate and one end point with the distance formula to find the radius • Plug center point and radius into equation for circle

Slope of Tangent • Slope of radius = Rise over Run (ΔY ÷ ΔX) • Find negative reciprocal – Change sign, flip fraction • Insert negative reciprocal into slope formula – Y = mx + b – Substitute y & x coords from tangent point to find b • Rewrite equation with y & x and the b value

Example: Writing the Equation of a Tangent Write the equation of the line tangent to the circle x 2 + y 2 = 29 at the point (2, 5). Step 1 Identify the center and radius of the circle. 2 2 From the equation x + y = 29, the circle has center of (0, 0) and radius r =.

Tangent Example Continued Step 2 Find the slope of the radius at the point of tangency and the slope of the tangent. Use the slope formula. Substitute (2, 5) for (x 2 , y 2 ) and (0, 0) for (x 1 , y 1 ). 5 The slope of the radius is 2. Because the slopes of perpendicular lines are negative reciprocals, the slope of the tangent is – 2. 5

Tangent Example Continued Step 3 Find the slope-intercept equation of the tangent by using the point (2, 5) and the slope m = –. 2 5 Use the point-slope formula. Substitute (2, 5) (x 1 , y 1 ) and – 2 for 5 Rewrite in slope-intercept form.

Tangent Example Continued The equation of the line that is tangent to 2 2 x + y = 29 at (2, 5) is Check Graph the circle and the line.

Examples with graph paper