What angle does an arc of 10 cm subtend at the centre of a circle of radius 10 cm? Give the answer to the nearest 0.1°.

Here are the solutions to the problems: Problem 7: In Fig. 12.32, O is the centre of the circle. The arcs are given in terms of 'a'. The total angle in a circle is 360^. The angles subtended by the arcs at the center are proportional to the length of the arcs. The sum of the arc lengths is 4a + a + 2a + 3a = 10a. This total arc length corresponds to 360^. So, 10a = 360^, which means a = 36^. a) AOB Step 1: The arc AB has length a. AOB = a = 36^ b) ADB Step 1: ADB is an angle subtended by arc AB at the circumference. The angle subtended by arc AB at the center is AOB = a. Step 2: The angle at the circumference is half the angle at the center subtended by the same arc. ADB = (1)/(2) AOB = (1)/(2) a = (1)/(2) (36^) = 18^ c) AOD Step 1: The arc AD has length 4a. AOD = 4a = 4 × 36^ = 144^ d) ACD Step 1: ACD is an angle subtended by arc AD at the circumference. The angle subtended by arc AD at the center is AOD = 4a. Step 2: The angle at the circumference is half the angle at the center subtended by the same arc. ACD = (1)/(2) AOD = (1)/(2) (4a) = 2a = 2 × 36^ = 72^ e) BCD Step 1: BCD is an angle subtended by arc BD at the circumference. The arc BD has length 2a + 3a = 5a. Step 2: The angle subtended by arc BD at the center is BOD = 5a = 5 × 36^ = 180^. Step 3: The angle at the circumference is half the angle at the center subtended by the same arc. BCD = (1)/(2) BOD = (1)/(2) (5a) = (5)/(2) a = (5)/(2) × 36^ = 90^ f) BDC Step 1: BDC is an angle subtended by arc BC at the circumference. The arc BC has length 2a. Step 2: The angle subtended by arc BC at the center is BOC = 2a = 2 × 36^ = 72^. Step 3: The angle at the circumference is half the angle at the center subtended by the same arc. BDC = (1)/(2) BOC = (1)/(2) (2a) = a = 36^ g) CBD Step 1: CBD is an angle subtended by arc CD at the circumference. The arc CD has length 3a. Step 2: The angle subtended by arc CD at the center is COD = 3a = 3 × 36^ = 108^. Step 3: The angle at the circumference is half the angle at the center subtended by the same arc. CBD = (1)/(2) COD = (1)/(2) (3a) = (3)/(2) a = (3)/(2) × 36^ = 54^ h) CAD Step 1: CAD is an angle subtended by arc CD at the circumference. The arc CD has length 3a. Step 2: The angle subtended by arc CD at the center is COD = 3a = 3 × 36^ = 108^. Step 3: The angle at the circumference is half the angle at the center subtended by the same arc. CAD = (1)/(2) COD = (1)/(2) (3a) = (3)/(2) a = (3)/(2) × 36^ = 54^ Final Answers for Problem 7: a) 36^ b) 18^ c) 144^ d) 72^ e) 90^ f) 36^ g) 54^ h) 54^ Problem 8: Four pencils of diameter 7 mm are held in a square formation by an elastic band. The diameter of each pencil is D = 7 mm. The radius is r = D/2 = 3.5 mm. Step 1: Identify the straight sections of the band. The band forms a square around the four pencils. The length of each side of this square is equal to two diameters. Length of one straight section = 2D = 2 × 7 mm = 14 mm. There are 4 such straight sections. Total length of straight sections = 4 × 14 mm = 56 mm. Step 2: Identify the curved sections of the band. At each corner, the band curves around a quarter of a pencil's circumference. There are 4 such quarter-circles. The total length of the curved sections is equal to the circumference of one full pencil. Circumference of one pencil = D = × 7 mm. Step 3: Calculate the total length of the band. Total length = (Total length of straight sections) + (Total length of curved sections) Total length = 56 mm + 7 mm Using ≈ 3.14159: Total length ≈ 56 + 7 × 3.14159 = 56 + 21.99113 ≈ 77.99113 mm If using ≈ (22)/(7): Total length = 56 + 7 × (22)/(7) = 56 + 22 = 78 mm Final Answer for Problem 8: The length of the band is 78 mm (using = (22)/(7)). Problem 9: The elastic band is used to hold seven pencils as shown in Fig. 12.34. The diameter of each pencil is 7 mm. Step 1: Identify the straight sections of the band. The arrangement forms a hexagon. There are 6 straight sections. Each straight section connects the centers of two adjacent pencils. The length of each straight section is equal to two radii, which is one diameter. Length of one straight section = D = 7 mm. Total length of straight sections = 6 × 7 mm = 42 mm. Step 2: Identify the curved sections of the band. At each of the 6 corners of the hexagon, the band curves around a pencil. The angle of each turn is 60^ (since it's a regular hexagon, the exterior angle is 360^/6 = 60^). So, each curved section is (60)/(360) = (1)/(6) of the circumference of a pencil. There are 6 such curved sections. Total length of curved sections = 6 × ((1)/(6) × D) = D = × 7 mm. Step 3: Calculate the total length of the band. Total length = (Total length of straight sections) + (Total length of curved sections) Total length = 42 mm + 7 mm Using ≈ 3.14159: Total length ≈ 42 + 7 × 3.14159 = 42 + 21.99113 ≈ 63.99113 mm If using ≈ (22)/(7): Total length = 42 + 7 × (22)/(7) = 42 + 22 = 64 mm Final Answer for Problem 9: The length of the band is 64 mm (using = (22)/(7)). Problem 10: A piece of wire 22 cm long is bent into an arc of a circle of radius 4 cm. What angle does the wire subtend at the centre of the circle? Step 1: Identify the given values. Arc length s = 22 cm. Radius r = 4 cm. Step 2: Use the formula for arc length. The formula for arc length is s = r, where is the angle in radians. 22 = 4 Step 3: Solve for in radians. = (22)/(4) = 5.5 radians Step 4: Convert the angle to degrees. To convert radians to degrees, multiply by (180^)/(). = 5.5 × (180^)/() Using ≈ (22)/(7): = 5.5 × (180^)/(22/7) = (11)/(2) × (180^ × 7)/(22) = (1)/(2) × (180^ × 7)/(2) = (180^ × 7)/(4) = 45^ × 7 = 315^ Final Answer for Problem 10: The angle subtended at the center is 315^. Problem 11: What angle does an arc of 10 cm subtend at the centre of a circle of radius 10 cm? Give the answer to the nearest 0.1^. Step 1: Identify the given values. Arc length s = 10 cm. Radius r = 10 cm. Step 2: Use the formula for arc length. The formula for arc length is s = r, where is the angle in radians. 10 = 10 Step 3: Solve for in radians. = (10)/(10) = 1 radian Step 4: Convert the angle to degrees and round to the nearest 0.1^. To convert radians to degrees, multiply by (180^)/(). = 1 × (180^)/() Using ≈ 3.14159265: ≈ (180^)/(3.14159265) ≈ 57.2957795^ Rounding to the nearest 0.1^: ≈ 57.3^ Final Answer for Problem 11: The angle subtended at the center is 57.3^. Drop the next question! 📸