answer:1. **Triangle and Circle Properties:** triangle ABC being isosceles with each of angle BAC and angle BCA as 45^circ implies angle ABC = 90^circ (sum of angles in a triangle = 180^circ). Tangent lines overline{BA} and overline{BC} ensure that angle BAO = angle BCO = 90^circ. 2. **Angle Analysis:** With overline{BA} and overline{BC} as tangents, angle OAC = angle OCA = 22.5^circ, since the tangent to a circle forms a 90^circ angle with the radius and splits the base angles of the isosceles triangle. Therefore, angle AOC = 180^circ - 22.5^circ - 22.5^circ = 135^circ. 3. **Determining Segment Relations:** Since overline{BO} includes both BD and DO, and DO = OA = BO cdot sin(22.5^circ). For simplicity, recall sin(22.5^circ) = frac{sqrt{2 - sqrt{2}}}{2}. 4. **Calculating BD:** BD = BO - DO = BO - BO cdot frac{sqrt{2 - sqrt{2}}}{2}. So, BD = BO cdot left(1 - frac{sqrt{2 - sqrt{2}}}{2}right). 5. **Compute the Ratio frac{BD}{BO}:** [ frac{BD}{BO} = 1 - frac{sqrt{2 - sqrt{2}}}{2} ] Thus, the ratio frac{BD}{BO} is 1 - frac{sqrt{2 - sqrt{2}}{2}}. The final answer is boxed{text{(A) } 1 - frac{sqrt{2 - sqrt{2}}}{2}}

answer:The circumference of a circle is given by C = 2pi r. Substituting the given radius, we have: C = 2 times 3.14 times 4 C = 25.12 (centimeters) The area of a circle is given by A = pi r^2. Substituting the given radius, we have: A = 3.14 times 4^2 A = 50.24 (square centimeters) So, the circumference of the coaster is boxed{25.12} centimeters, and the area is boxed{50.24} square centimeters. You can use the formulas for the circumference of a circle (C = 2pi r) and the area of a circle (A = pi r^2) to calculate the answers. This problem primarily tests your understanding and application of the formulas for the circumference and area of a circle.

answer:First, calculate the total number of ways to choose 4 articles of clothing out of the total. The total articles are 6 + 7 + 8 + 3 = 24 articles. - Compute total combinations to choose 4 out of 24: [ binom{24}{4} = frac{24 times 23 times 22 times 21}{4 times 3 times 2 times 1} = 12,650 ] Next, calculate the ways to choose one of each type of clothing: - Ways to choose one shirt = 6 - Ways to choose one pair of shorts = 7 - Ways to choose one pair of socks = 8 - Ways to choose one hat = 3 Multiply these together to find the favorable outcomes: [ 6 times 7 times 8 times 3 = 1,008 ] Now, calculate the probability: [ frac{1,008}{12,650} = boxed{frac{144}{1815}} ] Conclusion: The probability of choosing one shirt, one pair of shorts, one pair of socks, and one hat when choosing four articles of clothing from the drawer is boxed{frac{144}{1815}}.

answer:This is an infinite geometric series where the first term ( a ) is 1, and the common ratio ( r ) is ( frac{1}{4} ). The sum of an infinite geometric series is given by the formula: [ S = frac{a}{1 - r} ] Substituting the values of ( a ) and ( r ) into the formula: [ S = frac{1}{1 - frac{1}{4}} = frac{1}{frac{3}{4}} = frac{4}{3} ] Thus, the sum of the series is ( boxed{frac{4}{3}} ).