answer:# Problem: ABNM is a quadrilateral with a right angle. Given are AB, AM, and AC (where C is a point on the line MN), and the angle ABA' = x. The segment A'B' = AB and point A' falls inside the quadrilateral ABNM. The line AB is horizontal and can be rotated around point B, supported by the horizontal line MN at line AC. If the line AB is lowered to be horizontal (thus forming an angle x), the line AC takes the position A'D, moving away from point C to D. What is the length of CD? To solve for the length of CD, we start by analyzing the problem and the given geometric transformations. 1. **Identify and Define Points and Lines:** - Let A'B' = AB with A' inside the quadrilateral. - When AB is rotated around point B by an angle x, A moves to A'. - Line AC, initially at C, moves to position A'D. 2. **Relate Coordinates and Transformations:** - Assume initial coordinates: ( A = (0, 0) ), ( B = (a, 0) ), where a is the length of AB. - Point M lies directly above A: ( M = (0, b) ), b being the length AM. - Point C lies somewhere on the line MN: ( C = (c, b) ). 3. **Apply Rotation to Point A:** - Rotation of A by angle x around B: New coordinates ( A' = left(a cos(x), a sin(x)right) ). - Since A'B = AB, the coordinates remain consistent considering any possible orientation. 4. **Determine Coordinates of D:** - The line A'C before movement converts to A'D after rotation. - Horizontal distance moved by C to D is CD. - ( A = (0, 0) rightarrow A' = (a cos(x), a sin(x)) ). 5. **Calculate CD:** - Initial coordinates of C: ( (c, b) ). New point D on this trajectory due to rotation. - Line AC rightarrow A'D: Reflect positions and calculate displacement. One calculation approach on transforming C movements horizontally due to rotation includes a geometric shift: [ CD = c cdot cos(x) ] 6. **Conclusion:** Detailed comparison and application of coordinate shift, leading to: [ CD = AC cdot cos(x) ] Using given assumptions, - CD simplifies functionally to the distance formula. Hence, the final distance moved, calculated, remains consistent by cosine decomposing the translational formulae. [ boxed{CD = AC cdot cos(x)} ]

answer:(1) From the equation of line l: frac{x}{m}+frac{y}{4-m}=1, we can see that line l passes through points (m, 0) and (0, 4-m). Thus, the slope k=frac{4-m}{-m}=2. Solving for m, we get m=-4. (2) Line l passes through points (m,0) and (0,4-m). From m > 0 and 4-m > 0, we get 0 < m < 4. The area of triangle AOB is given by S_{triangle AOB}=frac{m(4-m)}{2}=frac{-(m-2)^2+4}{2}. When m=2, S_{triangle AOB} has the maximum value of 2. At this time, the equation of line l is boxed{x+y-2=0}.

answer:Jungkook originally had 3 red balls. After Yoongi gave him 1 blue ball, Jungkook now has a total of 3 red balls + 1 blue ball = boxed{4} balls.

answer:For Maria's investment, the formula for compound interest is applicable: [ A = P(1 + r)^n ] where - ( P = 1,000 ) (initial principal), - ( r = 0.05 ) (annual interest rate), - ( n = 5 ) (number of years). Plugging in the values, we get: [ A = 1000(1 + 0.05)^5 = 1000(1.05)^5 ] To compute ( (1.05)^5 ): [ (1.05)^5 approx 1.27628 ] Thus, [ A = 1000 times 1.27628 = 1276.28 ] The total interest earned is: [ text{Interest} = A - P = 1276.28 - 1000 = boxed{276.28} ]