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Line WV is perpendicular to both line RS and line TU.

On a coordinate plane, 3 lines are shown. Line V W goes through (negative 1, 5) and (1, 5). Line R S goes through (negative 5, 3) and (5, 1). Line T U goes through (negative 1, negative 2) and (4, negative 3).

Which statement must be true about line TU?

Line TU is parallel to line RS.
Line TU is perpendicular to line RS.
Line TU has no slope.
Line TU has a slope of –5.

1 answer

To analyze the relationship between line TU and line RS, we first need to determine the slopes of both lines.

1. Calculate the slope of line RS: Line RS goes through the points \((-5, 3)\) and \( (5, 1)\).

The slope \(m\) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \((x_1, y_1) = (-5, 3)\) and \((x_2, y_2) = (5, 1)\).

\[ m_{RS} = \frac{1 - 3}{5 - (-5)} = \frac{-2}{10} = -\frac{1}{5} \]

2. Calculate the slope of line TU: Line TU goes through the points \((-1, -2)\) and \( (4, -3)\).

Using the same formula: \[ m_{TU} = \frac{-3 - (-2)}{4 - (-1)} = \frac{-3 + 2}{4 + 1} = \frac{-1}{5} \]

3. Relationship between the slopes:

  • The slope of line RS is \(-\frac{1}{5}\).
  • The slope of line TU is also \(-\frac{1}{5}\).

Since both slopes are equal, this means that line TU is parallel to line RS.

Considering the options provided:

  • Line TU is parallel to line RS. (True)
  • Line TU is perpendicular to line RS. (False)
  • Line TU has no slope. (False)
  • Line TU has a slope of –5. (False)

Thus, the statement that must be true about line TU is: Line TU is parallel to line RS.