Question

shannon says that the lines ( y = -3x - 4 ), ( y = \frac{1}{3}x + 6 ), ( y = -4x - 5 ), and ( y = \frac{1}{4}x - 5 ) could represent the sides of a rectangle. explain shannon’s error. in a rectangle, opposite sides are parallel and adjacent sides are perpendicular. since dropdown lines have the same slope, dropdown also, since dropdown lines have slopes with a product of -1.

Explanation:

Step1: Recall slope rules for parallel/perpendicular lines

For two lines with slopes \(m_1\) and \(m_2\):

• Parallel: \(m_1 = m_2\)
• Perpendicular: \(m_1 \cdot m_2 = -1\)

Step2: Identify slopes of given lines

• Line \(y = -3x - 4\): slope \(m_1 = -3\)
• Line \(y = \frac{1}{3}x + 6\): slope \(m_2 = \frac{1}{3}\)
• Line \(y = -4x - 5\): slope \(m_3 = -4\)
• Line \(y = \frac{1}{4}x - 5\): slope \(m_4 = \frac{1}{4}\)

Step3: Check parallel pairs

• \(m_1 = -3\) and \(m_3 = -4\): \( -3

eq -4\) (not parallel)

• \(m_2 = \frac{1}{3}\) and \(m_4 = \frac{1}{4}\): \(\frac{1}{3}

eq \frac{1}{4}\) (not parallel)

• Correct parallel pairs: \(m_1 = -3\) & \(m_3 = -4\) is wrong; actually, \(m_1 = -3\) and no other line has slope \(-3\), \(m_3 = -4\) and no other line has slope \(-4\). Similarly for \(m_2\) and \(m_4\).

Step4: Check perpendicular pairs

• \(m_1 \cdot m_2 = (-3) \cdot \frac{1}{3} = -1\) (perpendicular)
• \(m_3 \cdot m_4 = (-4) \cdot \frac{1}{4} = -1\) (perpendicular)

Step5: Analyze rectangle side requirements

A rectangle has two pairs of parallel sides (opposite sides) and adjacent sides perpendicular (slope product \(-1\)). Shannon’s lines:

• Parallel? \(y = -3x - 4\) (slope \(-3\)) has no parallel line (no other line with slope \(-3\)); \(y = -4x - 5\) (slope \(-4\)) has no parallel line.
• Perpendicular? \(y = -3x - 4\) & \(y = \frac{1}{3}x + 6\) are perpendicular, \(y = -4x - 5\) & \(y = \frac{1}{4}x - 5\) are perpendicular. But for a rectangle, we need two pairs of parallel lines (each pair with equal slope) and adjacent pairs perpendicular. Shannon’s set has no two lines with the same slope (so no parallel sides), so they can’t form a rectangle.

Answer:

Shannon’s error is that the lines do not form two pairs of parallel lines (no two lines have the same slope, so opposite sides are not parallel). For a rectangle, opposite sides must be parallel (equal slopes) and adjacent sides perpendicular (slope product \(-1\)). Here, \(y = -3x - 4\) (slope \(-3\)) has no parallel line, \(y = -4x - 5\) (slope \(-4\)) has no parallel line, \(y = \frac{1}{3}x + 6\) (slope \(\frac{1}{3}\)) has no parallel line, and \(y = \frac{1}{4}x - 5\) (slope \(\frac{1}{4}\)) has no parallel line. Thus, the lines cannot represent a rectangle’s sides.