6.12,080 in in m cm ft in m cm 2 GEOMETRY LESSON 7-5 Pages Exercises 1.m 1 = 120; m 2 = 60; m 3 = 30 2.m 4 = 90; m 5 = 45; m 6 = 45 3.m 7 = 60; m 8 = 30; m 9 = cm ft in ft m m in a.72 b a.45 b.67.5

21.a.40 b a.30 b ft 2 24.a.9.1 in. b.6 in. c.3.7 in. GEOMETRY LESSON (continued) d.Answers may vary. Sample: About 4 in.; the length of a side of a pentagon should be between 3.7 in. and 6 in. 25.m 1 = 36; m 2 = 18; m 3 = The apothem is one leg of a rt. and the radius is the hypotenuse cm in m ft cm 2 32.a–c. regular octagon

GEOMETRY LESSON (continued) d.Construct a 60° angle with vertex at circle’s center m 2 34.Check students’ work cm cm 2, 41.6 cm m 2, m (continued) b.apothem = ; A = ap = (3s) A = s 2 3 s s ft in. 2, 27.7 in m 2, 70.1 m 2 41.a.b = s; h = s A = bh A = s s A = s 2 3

7.6 Circles and Arcs Objective:  To find the measures of central angles and arcs  To find the circumference and arc length

Definitions Circle – the set of all points equidistant from a given point called the center Center of a Circle – the point from which all points are equidistant Radius – a segment that has one endpoint at the center and the other endpoint on the circle

Definitions Congruent Circles – circles that have congruent radii Diameter – a segment that contains the center of a circle and has both endpoints on the circle Central Angle – an angle whose vertex is the center of the circle

Definitions Circumference – the distance around the circle Pi ( ∏ ) – the ration of the circumference of a circle to its diameter

Examples What if we are given a pie chart that represents data that have been collected? How can we find the measure of the arc or the measure of the angle?

Because there are 360° in a circle, multiply each percent by 360 to find the measure of each central angle. 65+ : 25% of 360 = = 90 45–64: 40% of 260 = = –44: 27% of 360 = = 97.2 Under 25: 8% of 360 = = 28.8 GEOMETRY LESSON 7-6 Circles and Arcs A researcher surveyed 2000 members of a club to find their ages. The graph shows the survey results. Find the measure of each central angle in the circle graph.

Circles and Arcs Some info to really help  The measure of the arc is the same as the measure of the central angle which creates that arc

Arcs A semicircle is half of a circle T P S R T P S R T P S R A minor arc is smaller than a semicircle A major arc is greater than a semicircle

. GEOMETRY LESSON 7-6 Circles and Arcs Identify the minor arcs, major arcs, and semicircles in P with point A as an endpoint. Minor arcs are smaller than semicircles. Two minor arcs in the diagram have point A as an endpoint, AD and AE. Major arcs are larger than semicircles. Two major arcs in the diagram have point A as an endpoint, ADE and AED. Two semicircles in the diagram have point A as an endpoint, ADB and AEB.

Arcs Adjacent Arcs – arcs of the same circle that have exactly one point in common Congruent Arcs – arcs that have the same measure and are in the same circle or in congruent circles Concentric Circles – circles that lie in the same plane and have the same center Arc length – a fraction of a circle’s circumference

Postulate 7-1: Arc Addition Postulate The measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs. mABC = mAB + mBC

mDXM = Substitute. mDXM = 236Simplify. mXY = mXD + mDYArc Addition Postulate mXY = m XCD + mDYThe measure of a minor arc is the measure of its corresponding central angle. mXY = Substitute. mXY = 96Simplify. Find mXY and mDXM in C.. mDXM = mDX + mXWMArc Addition Postulate GEOMETRY LESSON 7-6 Circles and Arcs

Circumference The circumference of a circle is the product of pi ( ∏ ) and the diameter. C = ∏ d or C = 2 ∏ r

C = dFormula for the circumference of a circle C = (24)Substitute. A circular swimming pool with a 16-ft diameter will be enclosed in a circular fence 4 ft from the pool. What length of fencing material is needed? Round your answer to the next whole number. The pool and the fence are concentric circles. The diameter of the pool is 16 ft, so the diameter of the fence is = 24 ft. Use the formula for the circumference of a circle to find the length of fencing material needed. About 76 ft of fencing material is needed. Draw a diagram of the situation. C 3.14(24)Use 3.14 to approximate. C 75.36Simplify. GEOMETRY LESSON 7-6 Circles and Arcs 7-6

Arc Length The length of an arc of a circle is the product of the ratio measure of the arc 360 and the circumference of the circle Length of AB = mAB/360 *2 ∏ r

length of ADB = 2 (18)Substitute The length of ADB is 21 cm. Find the length of ADB in M in terms of.. GEOMETRY LESSON 7-6 Circles and Arcs length of ADB = mADB 360 length of ADB = 2 rArc Length Formula Because mAB = 150, mADB = 360 – 150 = 210.Arc Addition Postulate

1. A circle graph has a section marked “Potatoes: 28%.” What is the measure of the central angle of this section? 2. Explain how a major arc differs from a minor arc. Use O for Exercises 3–6. 3. Find mYW. 4. Find mWXS. 5. Suppose that P has a diameter 2 in. greater than the diameter of O. How much greater is its circumference? Leave your answer in terms of. 6. Find the length of XY. Leave your answer in terms of.... A major arc is greater than a semicircle. A minor arc is smaller than a semicircle GEOMETRY LESSON 7-6 Circles and Arcs

Assignment P  #1-32 odd, 34-39