Number 7 11:04 PM Wed Sep 21=Section 4 The CircleOur second theorem concerns two tangents to a circle drawn from an externalpoint, Fig. 6-68.Two TangentsDrawn to a CircleHere, PA equals PB. Also, angle APC equalsangle BPC.Turning now from tangents to chords, Fig. 6-69, we have the theoremIf two chords in a circle intersect, the product ofthe parts of one chord equals the product of the partsof the other chordHere,IntersectingChordsTwo tangents drawn to a circle from a pointoutside the circle make equal angles with a linedrawn from the circle's center to the externalpoint. The distances from the external pointto each point of tangency are equal.+++ Example 22: Find x in Fig. 6-70.Solution: By theorem 85 we can write刀ab = cd16.3x10.1(12.5)x =:10.1(12.5)16.3= 7.758485SemicircleAny angle inscribed in a semicircle is a right angle.+++ Example 23: Find the distance x in Fig. 6-71.Solution: By statement 82, we know that = 90°. Then, by the Pythagorean theorem,x=√(250)² (148)² = 20182***Exercise 4 The Circle1. Find the circumference and area of a circle of radius 4.82 cm.2. Find the circumference and area of a circle of radius 2.385 in.3. Find the radius and area for a circle whose circumference is 74.8 in.4. Find the radius and area for a circle whose circumference is 2.73 m.5. Find the radius and circumference of a circle whose area is 39.5 ft².6. Find the radius and circumference of a circle whose area is 2.74 m².7. Inscribed and Circumscribed. A polygon is said to be inscribed in a circle ifevery vertex of the polygon lies on the circle. The circle is then said to be cir-cumscribed about the polygon. A polygon is said to be circumscribed about acircle if the circle is tangent to each side of the polygon. The circle is then saidto be inscribed in the polygon.8+1QAA ĘFIGURE 6-68aFIGURE 6-6916.312.5FIGURE 6-700FIGURE 6-71A193/1040B10.1193bSemicircle of radius = 125x91% 11:04 PM Wed Sep 21=194ΣA square circumscribedabout a circleA squareinscribedin a circleFIGURE 6-7248.0 cm4.12FIGURE 6-7352.0 cmFIGURE 6-763.16:Chapter 6 GeometryHow much larger is the side of a square circumscribing a circle 155 cm indiameter than a square inscribed in the same circle, Fig. 6–72?8. The radius of a circle is 5.00 m. Find the diameter of another circle containing4 times the area of the first.9. Find the distance x in Fig. 6-73.10. In Fig. 6-74, distance OP is 8.65 units. Find the distance PQ.11. Figure 6-75 shows a semicircle with a diameter of 105. Find distance PQ.Applications12. In a park is a circular fountain whose basin is 22.5 m in circumference. Whatis the diameter of the basin?13. The area of the bottom of a circular pan is 196 in.². What is its diameter?14. Find the diameter of a circular solar pond that has an area of 125 m².15. What is the circumference of a circular lake 33.0 m in diameter?16. The distance around a circular park is 2.50 mi. How many acres does it contain?(1 sq. mi = 640 acres)17. A woodcutter uses a tape measure and finds the circumference of a tree to be95.0 in. Assuming the tree cross-section to be circular, what length of chain-saw bar is needed to fell the tree? (By cutting from both sides, one can fell atree whose diameter is twice the length of the chain-saw bar.)18. What must be the diameter d of a cylindrical piston so that a pressure of125 lb/in.² on its circular end will result in a total force of 3610 lb? Hint: The fa surface is equal to the area of that surface times the pressure on that surface.6.35 diameterQAA19. Find the length of belt needed to connect the three 15.0-cm-diameter pulleysin Fig. 6-76. Hint: The total curved portion of the belt is equal to the circum-ference of one pulley.FIGURE 6-74FIGURE 6-7760°Q1D95.0FIGURE 6-75T= 1.500 in.0.1000 in. diameterFIGURE 6-78 Measuring a screwthread "over wires."194/1040R91%

Question

Number 7 11:04 PM Wed Sep 21=Section 4 The CircleOur second theorem concerns two tangents to a circle drawn from an externalpoint, Fig. 6-68.Two TangentsDrawn to a CircleHere, PA equals PB. Also, angle APC equalsangle BPC.Turning now from tangents to chords, Fig. 6-69, we have the theoremIf two chords in a circle intersect, the product ofthe parts of one chord equals the product of the partsof the other chordHere,IntersectingChordsTwo tangents drawn to a circle from a pointoutside the circle make equal angles with a linedrawn from the circle's center to the externalpoint. The distances from the external pointto each point of tangency are equal.+++ Example 22: Find x in Fig. 6-70.Solution: By theorem 85 we can write刀ab = cd16.3x10.1(12.5)x =:10.1(12.5)16.3= 7.758485SemicircleAny angle inscribed in a semicircle is a right angle.+++ Example 23: Find the distance x in Fig. 6-71.Solution: By statement 82, we know that = 90°. Then, by the Pythagorean theorem,x=√(250)² (148)² = 20182***Exercise 4 The Circle1. Find the circumference and area of a circle of radius 4.82 cm.2. Find the circumference and area of a circle of radius 2.385 in.3. Find the radius and area for a circle whose circumference is 74.8 in.4. Find the radius and area for a circle whose circumference is 2.73 m.5. Find the radius and circumference of a circle whose area is 39.5 ft².6. Find the radius and circumference of a circle whose area is 2.74 m².7. Inscribed and Circumscribed. A polygon is said to be inscribed in a circle ifevery vertex of the polygon lies on the circle. The circle is then said to be cir-cumscribed about the polygon. A polygon is said to be circumscribed about acircle if the circle is tangent to each side of the polygon. The circle is then saidto be inscribed in the polygon.8+1QAA ĘFIGURE 6-68aFIGURE 6-6916.312.5FIGURE 6-700FIGURE 6-71A193/1040B10.1193bSemicircle of radius = 125x91% 11:04 PM Wed Sep 21=194ΣA square circumscribedabout a circleA squareinscribedin a circleFIGURE 6-7248.0 cm4.12FIGURE 6-7352.0 cmFIGURE 6-763.16:Chapter 6 GeometryHow much larger is the side of a square circumscribing a circle 155 cm indiameter than a square inscribed in the same circle, Fig. 6–72?8. The radius of a circle is 5.00 m. Find the diameter of another circle containing4 times the area of the first.9. Find the distance x in Fig. 6-73.10. In Fig. 6-74, distance OP is 8.65 units. Find the distance PQ.11. Figure 6-75 shows a semicircle with a diameter of 105. Find distance PQ.Applications12. In a park is a circular fountain whose basin is 22.5 m in circumference. Whatis the diameter of the basin?13. The area of the bottom of a circular pan is 196 in.². What is its diameter?14. Find the diameter of a circular solar pond that has an area of 125 m².15. What is the circumference of a circular lake 33.0 m in diameter?16. The distance around a circular park is 2.50 mi. How many acres does it contain?(1 sq. mi = 640 acres)17. A woodcutter uses a tape measure and finds the circumference of a tree to be95.0 in. Assuming the tree cross-section to be circular, what length of chain-saw bar is needed to fell the tree? (By cutting from both sides, one can fell atree whose diameter is twice the length of the chain-saw bar.)18. What must be the diameter d of a cylindrical piston so that a pressure of125 lb/in.² on its circular end will result in a total force of 3610 lb? Hint: The fa surface is equal to the area of that surface times the pressure on that surface.6.35 diameterQAA19. Find the length of belt needed to connect the three 15.0-cm-diameter pulleysin Fig. 6-76. Hint: The total curved portion of the belt is equal to the circum-ference of one pulley.FIGURE 6-74FIGURE 6-7760°Q1D95.0FIGURE 6-75T= 1.500 in.0.1000 in. diameterFIGURE 6-78 Measuring a screwthread &#34;over wires.&#34;194/1040R91%

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How much larger is the side of a square circumscribing a circle 155 cm in diameter than a square inscribed in the same circle, Fig. 6-72? The rading of a circle is 500 m. Find the diameter of another circle containing