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Difficulty: MediumLinear Equations and Graphing

In the standard (x,y)(x, y) coordinate plane, the line 3x−4y=243x - 4y = 24 intersects the xx-axis at point AA and the yy-axis at point BB. What is the length of the segment ABAB?

  1. 1010Answer
  2. B
    272\sqrt{7}
  3. C
    1414
  4. D
    22
  5. E
    43\frac{4}{3}

Answer

The length of the segment ABAB is 1010.
To find the length of segment ABAB, we first determine the coordinates of points AA and BB. The xx-intercept, AA, is found by setting y=0y = 0, giving 3x=24⇒x=83x = 24 \Rightarrow x = 8, so A=(8,0)A = (8, 0). The yy-intercept, BB, is found by setting x=0x = 0, giving −4y=24⇒y=−6-4y = 24 \Rightarrow y = -6, so B=(0,−6)B = (0, -6). Using the distance formula, the distance between (8,0)(8, 0) and (0,−6)(0, -6) is (8−0)2+(0−(−6))2=64+36=100=10\sqrt{(8 - 0)^2 + (0 - (-6))^2} = \sqrt{64 + 36} = \sqrt{100} = 10. This matches the correct option.

Step-by-Step Solution

1
Find the coordinates of point AA, the xx-intercept of the line.
Point AA is (8,0)(8, 0).
Set y=0y = 0 in the equation 3x−4y=243x - 4y = 24, which gives 3x=243x = 24, so x=8x = 8.
2
Find the coordinates of point BB, the yy-intercept of the line.
Point BB is (0,−6)(0, -6).
Set x=0x = 0 in the equation 3x−4y=243x - 4y = 24, which gives −4y=24-4y = 24, so y=−6y = -6.
3
Calculate the distance between point A(8,0)A(8, 0) and point B(0,−6)B(0, -6) using the distance formula.
The distance is 1010.
The distance formula is d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}. Substituting the coordinates gives d=(0−8)2+(−6−0)2=(−8)2+(−6)2=64+36=100=10d = \sqrt{(0 - 8)^2 + (-6 - 0)^2} = \sqrt{(-8)^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10.

Key Concept

Finding intercepts of a linear equation and calculating the distance between two points on the coordinate plane.
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Topics

Linear Equations and Graphing

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