answer:1. Substitute the point (5, 15) into the first line equation y = nx + 5: [ 15 = 5n + 5 quad Rightarrow quad 10 = 5n quad Rightarrow quad n = 2 ] 2. Substitute the point (5, 15) into the second line equation y = 4x + c: [ 15 = 4 times 5 + c quad Rightarrow quad 15 = 20 + c quad Rightarrow quad c = -5 ] 3. Calculate c + n: [ c + n = -5 + 2 = boxed{-3} ]

answer:Setting up the diagram: [asy] pair A,B,C; B = (0,0); C = (12,0); A = (0,12*Tan(50)); draw(A--B--C--A); draw(rightanglemark(C,B,A,18)); label("B",B,SW); label("C",C,SE); label("A",A,N); label("12",C/2,S); [/asy] To find AB, we make use of the tangent of angle A in right triangle ABC: [ tan A = frac{BC}{AB} ] Given tan A = tan(40^circ) and the length of BC = 12, we rearrange to find AB: [ AB = frac{BC}{tan A} = frac{12}{tan(40^circ)} ] Thus, [ AB approx frac{12}{0.8391} approx boxed{14.3} ] This uses a calculator value of tan(40^circ). Conclusion: boxed{14.3} is the calculated length of side AB to the nearest tenth.

answer:To address the propositions given in the problem, we analyze them based on geometric theorems and properties. **A:** The statement claims if line m is parallel to plane alpha, and line n is also parallel to plane alpha, then line m must be parallel to line n. This is not necessarily true because, while both m and n do not intersect plane alpha, it does not imply that m and n are parallel to each other; they could be skew lines (lines that do not intersect and are not parallel), or they could even intersect outside of plane alpha. Hence, statement A is incorrect. **B:** This statement says if lines m and n are both contained in plane alpha and are both parallel to plane beta, then alpha is parallel to beta. The missing condition here is whether line m and line n intersect. Without additional information on the relationship between m and n, or a more direct relationship between alpha and beta, we cannot conclude that alpha is parallel to beta. Thus, statement B is incorrect. **C:** The proposition indicates that if plane alpha is perpendicular to plane beta, and line m is contained in plane alpha, then m is perpendicular to beta. This is not automatically true because, although alpha is perpendicular to beta, line m could be oriented in any direction within alpha and thus does not necessarily intersect beta perpendicularly. The theorem of perpendicular planes does not guarantee that every line within alpha is perpendicular to beta. Therefore, statement C is incorrect. **D:** Finally, this statement asserts if plane alpha is perpendicular to plane beta, line m is perpendicular to beta, and line m is not contained in plane alpha, then m is parallel to alpha. Given that m is perpendicular to beta and not contained in alpha, the only logical spatial relationship left is for m to be parallel to alpha. This is because the perpendicularity to beta dictates a specific orientation of m that, not intersecting alpha, can only mean it runs parallel to alpha without ever intersecting it. Therefore, the correct choice, following the analysis of each statement, is (boxed{D}).

answer:First, we find the diagonal of the square. The diagonal splits the square into two 45-45-90 triangles. The length of the diagonal (hypotenuse) in a 45-45-90 triangle is (sqrt{2}) times the length of a leg (the side of the square). Here, the side of the square is 70sqrt{2} cm. Thus, the diagonal is: [ 70sqrt{2} times sqrt{2} = 70(sqrt{2})^2 = 70 times 2 = 140 text{ cm}. ] Next, we calculate the circumference of the circle. The circle's diameter is the diagonal of the square, which is 140 cm. The circumference of a circle is given by (pi times text{diameter}): [ text{Circumference} = pi times 140 = 140pi text{ cm}. ] So, the final answers are: - The length of the diagonal of the square: (boxed{140 text{ cm}}). - The circumference of the circle: (boxed{140pi text{ cm}}).