answer:Let's consider a quadrilateral ABCD inscribed in a circle. We need to find the sum of the angles inscribed in the four segments outside the quadrilateral. 1. **Identify the Segments and Angles**: - The four segments outside the quadrilateral are the regions outside ABCD but inside the circle. - Let alpha, beta, gamma, and delta be the angles inscribed in the segments outside ABCD near vertices A, B, C, and D respectively. 2. **Angle Properties**: - Each angle inscribed in a segment measures half the degree measure of the arc it subtends. - The total degree measure of the circle is 360^circ. 3. **Arcs Subtended by Quadrilateral Angles**: - The angle at vertex A of the quadrilateral subtends an arc equal to 360^circ - text{arc } BC. - Similarly, the angles at vertices B, C, and D subtend arcs 360^circ - text{arc } CD, 360^circ - text{arc } DA, and 360^circ - text{arc } AB respectively. 4. **Sum of Angles in Segments**: - The angle alpha in the segment near A subtends the arc BC, so alpha = frac{1}{2}(360^circ - text{arc } BC). - Similarly, beta = frac{1}{2}(360^circ - text{arc } CD), gamma = frac{1}{2}(360^circ - text{arc } DA), and delta = frac{1}{2}(360^circ - text{arc } AB). 5. **Calculate the Total Sum**: - The sum of these angles is: [ alpha + beta + gamma + delta = frac{1}{2}[(360^circ - text{arc } BC) + (360^circ - text{arc } CD) + (360^circ - text{arc } DA) + (360^circ - text{arc } AB)] ] - Since the sum of the arcs BC, CD, DA, and AB is 360^circ, the equation simplifies to: [ alpha + beta + gamma + delta = frac{1}{2}[4 times 360^circ - 360^circ] = frac{1}{2} times 1080^circ = 540^circ ] Thus, the sum of the four angles inscribed in the segments outside the quadrilateral is boxed{540^circ}, which corresponds to choice textbf{(D)} 540.

answer:Since the equation of the circle is x^2+y^2-2x+6y+8=0, the coordinates of the center of the circle are (1, -3). Substituting into the options, we can see that option C is correct. Therefore, the answer is boxed{text{C}}.

answer:1. Determine the total edge length contribution from the six triangles. - Since each triangle shares one side with the hexagon, the effective contribution for each triangle is only two sides. - Hence, for six triangles, the total contribution to the perimeter is (2 times 6) sides. [ text{Edge length from triangles} = 2 text{(sides/triangle)} times 6 text{(triangles)} = 12 text{cm} ] 2. Determine the total edge length contribution from the six squares. - Each square has four sides, but since they share sides with adjacent squares and triangles, their effective contribution is lower. - Given that squares are arranged in a way that contributes two sides per square to the perimeter. - Hence, for six squares, the total contribution to the perimeter is (2 times 6) sides. [ text{Edge length from squares} = 2 text{(sides/square)} times 6 text{(squares)} = 12 text{cm} ] 3. Adding these contributions together gives the total perimeter. [ text{Total perimeter} = text{Edge length from triangles} + text{Edge length from squares} ] [ text{Total perimeter} = 12 text{cm} + 12 text{cm} = 24 text{cm} ] 4. Conclusion: The perimeter of the given figure is (boxed{24 text{cm}}).

answer:Given (0<a<b<c<d<300) and the equations: [ a + d = b + c, ] [ bc - ad = 91. ] We will determine the number of ordered quadruples ((a, b, c, d)) that satisfy these conditions. 1. **Express (a, b, c,) and (d) in terms of (a, x, y)**: Suppose: [ b = a + x, ] [ c = a + y, ] [ d = a + x + y, ] where (x) and (y) are positive integers with (x neq y). 2. **Plug these expressions into the given equations**: The first equation becomes: [ a + (a + x + y) = (a + x) + (a + y), ] which simplifies to: [ 2a + x + y = 2a + x + y. ] This identity checks out and does not provide new information. The second equation, (bc - ad = 91), becomes: [ (a + x)(a + y) - a(a + x + y) = 91. ] Simplifying the left-hand side: [ a^2 + ay + ax + xy - a^2 - ax - ay = xy. ] Hence: [ xy = 91. ] 3. **Find the factor pairs of 91**: Since (91 = 7 times 13), the factor pairs ((x, y)) are: [ (x, y) = (1, 91), (7, 13). ] 4. **Calculate the possible range of (a)**: For the pair ((1, 91)): [ b = a + 1, ] [ c = a + 91, ] [ d = a + 92. ] Given that (0 < a < 300), we have: [ a + 92 < 300 implies a < 208. ] So, (a) can range from 1 to 207, inclusive. There are 207 possible values for (a). For the pair ((7, 13)): [ b = a + 7, ] [ c = a + 13, ] [ d = a + 20. ] Given that (0 < a < 300), we have: [ a + 20 < 300 implies a < 280. ] So, (a) can range from 1 to 279, inclusive. There are 279 possible values for (a). 5. **Sum the possible values**: The total number of valid quadruples is: [ 207 + 279 = 486. ] # Conclusion: [ boxed{486} ]