answer:Start by simplifying (x-3)(x+3) using the difference of squares formula. [ (x-3)(x+3) = x^2 - 9 ] Next, multiply the result by (x^2 + 9). [ (x^2 - 9)(x^2 + 9) = x^4 - 81 ] Thus, the expanded form of (x-3)(x+3)(x^2+9) is: [ boxed{x^4 - 81} ]

answer:Solution: tan(-240^circ) = -tan(240^circ) = -tan(180^circ + 60^circ) = -tan(60^circ) = -sqrt{3}. Hence, the answer is boxed{-sqrt{3}}. By directly using the trigonometric identity (co-function identity), we can simplify and solve the problem. This question tests the application of trigonometric identities and the simplification of trigonometric functions, which is a basic knowledge assessment.

answer:To show that the arcs between two parallel chords of a circle are equal, we proceed as follows: 1. **Initial Setup**: Consider a circle with center ( O ). Let the chords ( AB ) and ( CD ) be parallel. 2. **Connecting Opposite Ends**: Draw the line segments ( AC ) and ( BD ). Since ( AB parallel CD ), we know that ( AC ) and ( BD ) intersect at some point ( P ), which is not necessarily the center of the circle ( O ). 3. **Identify Angles**: When you connect these endpoints, you form two pairs of vertically opposite angles at the point of intersection ( P ). 4. **Equality of Opposite Angles**: By the property of vertically opposite angles, we know: [ angle APC = angle BPD ] since ( AB parallel CD ) and each set of opposite angles formed where ( AC ) and ( BD ) intersect are equal. 5. **Arcs Subtended by Equal Angles**: In a circle, equal angles subtended by chords at any point on the circle ensure that the arcs that they subtend are equal in measure. Thus, the angle ( angle APC ) subtends an arc, say arc ( AC ), and ( angle BPD ) subtends arc ( BD ). 6. **Conclusion about the Arcs**: Since ( angle APC = angle BPD ), it follows that: [ text{Arc } AC = text{Arc } BD ] Hence, the arcs of the circle that are intercepted between the two chords are equal in length. [ boxed{} ]

answer:1. **Identify the Coordinates of the Triangle Vertices**: The triangle is formed by the points (-a, 0), (b, 0), and (0, h). The base of the triangle along the x-axis is b+a, and the height is along the y-axis h. 2. **Calculate the Height Using the Area Formula**: The area T of the triangle is given by: [ T = frac{1}{2} times text{base} times text{height} = frac{1}{2} times (b+a) times h ] Solving for h, we get: [ h = frac{2T}{b+a} ] 3. **Determine the Slope of the Line**: The line passes through the points (-a, 0) and (b, 0). The slope m of the line is calculated as: [ m = frac{h - 0}{b - (-a)} = frac{frac{2T}{b+a}}{b + a} = frac{2T}{(b+a)^2} ] 4. **Write the Equation of the Line in Slope-Intercept Form**: Using the slope m and y-intercept b = h = frac{2T}{b+a}, the equation of the line is: [ y = mx + b = frac{2T}{(b+a)^2}x + frac{2T}{b+a} ] 5. **Convert the Equation to Standard Form**: Multiplying through by (b+a)^2 to clear the fractions and rearranging, we get: [ (b+a)^2y = 2Tx + 2T(b+a) ] [ 0 = 2Tx - (b+a)^2y + 2T(b+a) ] The final equation of the line is: [ 2Tx - (b+a)^2y + 2T(b+a) = 0 ] The final answer is C) boxed{2Tx - (b+a)^2y + 2T(b+a) = 0}