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Line AA passes through the points (4,9)(4,9) and (12,23)(12,23). Line BB passes through the points (0,2)(0,2) and (−8,−12)(-8,-12). Which statement is true?\newlineChoices:\newline(A) Line AA intersects line BB at exactly one point.\newline(B) Line AA does not intersect line BB.\newline(C) Line AA overlaps line BB.

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Q. Line AA passes through the points (4,9)(4,9) and (12,23)(12,23). Line BB passes through the points (0,2)(0,2) and (−8,−12)(-8,-12). Which statement is true?\newlineChoices:\newline(A) Line AA intersects line BB at exactly one point.\newline(B) Line AA does not intersect line BB.\newline(C) Line AA overlaps line BB.
  1. Calculate Slope Line A: Calculate the slope of Line A using the formula (y2−y1)/(x2−x1)(y_2 - y_1) / (x_2 - x_1). For points (4,9)(4,9) and (12,23)(12,23):\newlineSlope of Line A = (23−9)/(12−4)=14/8=1.75(23 - 9) / (12 - 4) = 14 / 8 = 1.75.
  2. Calculate Slope Line B: Calculate the slope of Line B using the formula (y2−y1)/(x2−x1)(y_2 - y_1) / (x_2 - x_1). For points (0,2)(0,2) and (−8,−12)(-8,-12):\newlineSlope of Line B = (−12−2)/(−8−0)=−14/−8=1.75(-12 - 2) / (-8 - 0) = -14 / -8 = 1.75.
  3. Compare Slopes: Compare the slopes of Line extit{A} and Line extit{B}. Since both slopes are equal, the lines are either parallel or identical (overlapping). Next, check if they are overlapping by using the point-slope form of a line equation.
  4. Point-Slope Form Line A: Use the point-slope form y−y1=m(x−x1)y - y_1 = m(x - x_1) for Line A using point (4,9)(4,9) and slope 1.751.75:y−9=1.75(x−4)y - 9 = 1.75(x - 4)y=1.75x−7+9y = 1.75x - 7 + 9y=1.75x+2y = 1.75x + 2.
  5. Point-Slope Form Line B: Use the point-slope form y−y1=m(x−x1)y - y_1 = m(x - x_1) for Line B using point (0,2)(0,2) and slope 1.751.75:y−2=1.75(x−0)y - 2 = 1.75(x - 0)y=1.75x+2.y = 1.75x + 2.
  6. Identical Lines: Since both lines have the same equation y=1.75x+2y = 1.75x + 2, Line A and Line B are identical, meaning they overlap along their entire length.

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