answer:1. Consider a **regular polyhedron** with a center ( O ). These polyhedra include the five Platonic solids, each with identical faces and vertices distributed symmetrically. Denote one of its faces as ( F ) with centroid ( C_F ). 2. The crucial property of regular polyhedra is that they possess **symmetry axes through their center ( O )**, specifically, each axis passes through ( O ) and a vertex ( V ). 3. When we rotate the polyhedron around such an axis by **( frac{2pi}{m} )**, where ( m ) is the number of faces meeting at ( V ), the vertices and their adjacent faces align perfectly with one another. 4. Each face ( F ) of the regular polyhedron is a regular ( n )-gon, where ( n ) is the number of edges (and vertices) of the face. Thus, rotating around the symmetry axis involves mapping face centers to other face centers. 5. Due to the rotational symmetry of the polyhedron, the centroids ( C ) of faces must also align in a symmetric manner. Specifically, these centroids can be connected to form another polyhedron. 6. The polyhedron formed by connecting these face centroids, referred to as **dual polyhedra**, will also be regular. This newly formed regular polyhedron has vertices at the centers of the original faces and faces corresponding to the vertices of the original polyhedron. 7. This reciprocal relationship implies: the centers of faces of a regular polyhedron serve as the vertices of its dual polyhedron, and vice versa. 8. Some notable dual pairs of regular polyhedra are: - The Cube and the Octahedron. - The Dodecahedron and the Icosahedron. - The Tetrahedron, which is self-dual, meaning its dual is another tetrahedron. # Conclusion: Therefore, the centers of the faces of any regular polyhedron indeed form the vertices of another regular polyhedron. This principle exemplifies the concept of **duality in polyhedra**. Specifically, the centers of the faces of the second polyhedron correspond to the vertices of a third polyhedron that is geometrically similar (homothetic) to the original. Thus, the solution to the problem is: boxed{}

answer:According to the problem, we have vectors overrightarrow{a}=(2,4) and overrightarrow{b}=(-1,1). Therefore, overrightarrow{c} = overrightarrow{a} - t overrightarrow{b} = (2+t, 4-t). If overrightarrow{b} is perpendicular to overrightarrow{c}, this implies that their dot product equals zero: overrightarrow{b} cdot overrightarrow{c} = (-1)(2+t) + (1)(4-t) = -2 - t + 4 - t = 2 - 2t. Setting the dot product equal to zero, we solve for t: 2 - 2t = 0, 2t = 2, t = boxed{1}. Hence, the correct answer is B: 1. The problem assesses the ability to compute the dot product of vectors using their coordinates; the key is mastering the coordinate computation of the vector dot product.

answer:# Problem: Около окружности с диаметром 15 описана равнобедренная трапеция с боковой стороной, равной 17. Найдите основания трапеции. 1. **Identify Key Elements:** - The diameter of the inscribed circle (15 units) is also the height of the isosceles trapezoid since the height will be the perpendicular distance between the two parallel sides (bases). - Each leg (non-parallel side) of the trapezoid is 17 units. 2. **Label the trapezoid and inscribed circle:** - Let ABCD be the trapezoid with AB and CD as the bases, such that AB > CD. - Let h = 15 be the height (distance between AB and CD). 3. **Calculate the Segment from Vertices to the Base:** - Let M be the foot of the perpendicular from vertex C to AB. Since CM = h = 15, MD, a segment of base AB, can be calculated from the right triangle CMD: [ MD = sqrt{CD^2 - CM^2} ] The legs are equal, based on symmetry, M will split the line segment into equal lengths. 4. **Middle Line and Segments:** - Since the legs are equal, by symmetry, this length MD = 8 splits equally on both sides: [ MD = sqrt{17^2 - 15^2} = sqrt{289 - 225} = sqrt{64} = 8 ] Since this is half of the offset, each base segment contributes this value. 5. **Calculate Longer Base AD:** - Calculate the full longer base AD: [ AD = CD + 2 cdot MD = 17 + 2 cdot 8 = 17 + 16 = 25 ] 6. **Find the smaller base:** - By symmetry and diameter midpoint equation, AB criterion holds. Knowing the total sum of the segment, 7. **Final Single Step Calculation:** - Given AD + BC = 34): [ text{Since } 34 - 25 = 9. Thus, BC = 9 text{.} ] 8. **Conclusion**: - Hence, the two bases are (9) and (25), written as usually maximal to minimal: [ boxed{25, 9} ]

answer:Let's denote the number of turtles Kristen has as K. According to the information given, Kris has one fourth the number of turtles Kristen has, so Kris has K/4 turtles. Trey has 5 times as many turtles as Kris, so Trey has 5 * (K/4) turtles, which simplifies to (5K/4) turtles. The total number of turtles is the sum of Kristen's, Kris's, and Trey's turtles, which is given as 30. So we can write the equation: K + K/4 + 5K/4 = 30 Combining like terms, we get: K + (1/4)K + (5/4)K = 30 (1 + 1/4 + 5/4)K = 30 (1 + 6/4)K = 30 (4/4 + 6/4)K = 30 (10/4)K = 30 (5/2)K = 30 Now, to find K, we multiply both sides of the equation by the reciprocal of (5/2), which is (2/5): K = 30 * (2/5) K = 60/5 K = 12 So, Kristen has 12 turtles. Now we can find out how many turtles Kris and Trey have: Kris has K/4 turtles, so Kris has 12/4 = 3 turtles. Trey has 5K/4 turtles, so Trey has 5*12/4 = 5*3 = 15 turtles. The total number of turtles is the sum of Kristen's, Kris's, and Trey's turtles: Kristen's turtles + Kris's turtles + Trey's turtles = Total turtles 12 + 3 + 15 = 30 So, there are boxed{30} turtles altogether, which matches the given total.