answer:Given that A - B = C, D = 2(A + B), E = A cdot B, and F = 3 meters, Therefore, A = 28 meters, B = 15 meters, C = 13 meters, D = 86 meters, E = 420 square meters, and F = 3 meters. Thus, the answers are: A = boxed{28} meters, B = boxed{15} meters, C = boxed{13} meters, D = boxed{86} meters, E = boxed{420} square meters, and F = boxed{3} meters.

answer:1. Let us consider two skew lines with segments of lengths (a) and (b) that slide along these lines. These two lines intersect at an angle (varphi) and the perpendicular distance between these lines is (d). 2. We need to show that the volume of the tetrahedron formed by the endpoints of these segments does not depend on their particular positions along the skew lines. 3. The volume (V) of a tetrahedron given by vertices (A, B, C, D) is determined by the scalar triple product of vectors (overrightarrow{AB}), (overrightarrow{AC}), and (overrightarrow{AD}): [ V = frac{1}{6} left| overrightarrow{AB} cdot (overrightarrow{AC} times overrightarrow{AD}) right|. ] 4. Arrange the tetrahedron (ABCD) such that (A) and (B) are on the first line with distance (a) between them, while (C) and (D) are on the second line with distance (b) between them. The perpendicular distance (d) is between these two lines and the angle between them is (varphi). 5. To understand the volume, we first construct a parallelepiped using these lines. The planes containing these lines form the parallelepiped, with tetrahedra corresponding to the (ABCD) vertices as a subset: - The overall volume of a parallelepiped is given by the magnitude of the scalar triple product of its edge-forming vectors. 6. For segments (a) and (b), the volume of a parallelepiped formed by vectors ( overrightarrow{AB}) (length (a)), ( overrightarrow{AC}) (length (b)), and ( overrightarrow{d} ) (distance (d)) where ( varphi ) is angle between (a) and (b): [ text{Volume of parallelepiped} = V_p = a b d sin (varphi). ] 7. Since we know the tetrahedron is one of the subdivisions of the parallelepiped: [ V_{text{tetrahedron}} = frac{1}{6} V_p = frac{1}{6} a b d sin (varphi). ] 8. Now, examining the tetrahedron's structure: - It includes one-fourth of the total volume of the parallelepiped. - The total sum of the tetrahedra within the parallelepiped includes this particular configuration, maintaining the same proportional volume shares. Therefore, the volume of the tetrahedron formed by segments sliding on skew lines remains constant and is given by: [ V_{text{tetrahedron}} = frac{1}{6} a b d sin (varphi). ] # Conclusion: [ boxed{frac{1}{6} a b d sin (varphi)} ]

answer:1. Define the center of the inscribed circle as C, and let D be the point where the inscribed circle is tangent to arc AB. Let E and F be the points where the inscribed circle is tangent to OA and OB, respectively. 2. Since angles CEO, CFO, and EOF are all right angles, angle FCE is a right angle. Therefore, the measure of angle DCE is (360-90)/2 = 135 degrees. 3. By symmetry, angles ECO and FCO are congruent, each measuring (180-135)/2 = 22.5 degrees for a third of a circle sector. Therefore, angle DCO measures 135+22.5 = 157.5 degrees. 4. Since DC = r, and CO = rsqrt{2} (as triangle CEO is an isosceles right triangle), and OD, a radius of the circle centered at O, is 6 cm, we have DC+CO = r + rsqrt{2} rightarrow OD = 6 cm. 5. Solving for r, set r + rsqrt{2} = 6 cm, and solve for r using the equation: [ r(1 + sqrt{2}) = 6 Rightarrow r = frac{6}{1+sqrt{2}} cdot frac{sqrt{2}-1}{sqrt{2}-1} = 6(sqrt{2}-1) Rightarrow r = boxed{6sqrt{2}-6} text{ centimeters}. ]

answer:To find the area of a trapezium (trapezoid), we can use the formula: Area = (1/2) * (sum of the lengths of the parallel sides) * (distance between the parallel sides) However, in this case, we have an additional piece of information: the angle between the longer parallel side and the distance between the parallel sides is 30 degrees. This angle can be used to find the height of the trapezium if it is not perpendicular to the parallel sides. Since the distance between the parallel sides is given as 13 cm, and we have an angle of 30 degrees, we can calculate the actual height (h) of the trapezium using trigonometry. The height will be the side opposite the 30-degree angle in the right-angled triangle formed by the height, the distance between the parallel sides, and the slanted side of the trapezium. Using the sine function: sin(30 degrees) = opposite / hypotenuse sin(30 degrees) = h / 13 cm We know that sin(30 degrees) = 1/2, so: 1/2 = h / 13 cm h = (1/2) * 13 cm h = 6.5 cm Now that we have the height, we can use the area formula for a trapezium: Area = (1/2) * (sum of the lengths of the parallel sides) * (height) Area = (1/2) * (20 cm + 18 cm) * 6.5 cm Area = (1/2) * 38 cm * 6.5 cm Area = 19 cm * 6.5 cm Area = 123.5 cm² The area of the trapezium is boxed{123.5} square centimeters.