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 shannons error. in a rectangle, opposite sides are parallel and adjacent sides are perpendicular. since ( square ) lines have the same slope, ( square ). also, since ( square ) lines have slopes with a product of ( - 1 ), ( square )

Explanation:

Step1: Check for parallel lines

Parallel lines have the same slope. The slopes of the given lines are \(m_1=-3\), \(m_2 =-\frac{1}{3}\), \(m_3=-4\), \(m_4=\frac{1}{4}\). No two of these slopes are equal. So, there are no parallel lines among them.

Step2: Check for perpendicular lines

Perpendicular lines have slopes whose product is \(- 1\).

• \(m_1\times m_2=(-3)\times(-\frac{1}{3}) = 1

eq - 1\)

• \(m_1\times m_3=(-3)\times(-4)=12

eq - 1\)

• \(m_1\times m_4=(-3)\times\frac{1}{4}=-\frac{3}{4}

eq - 1\)

• \(m_2\times m_3=(-\frac{1}{3})\times(-4)=\frac{4}{3}

eq - 1\)

• \(m_2\times m_4=(-\frac{1}{3})\times\frac{1}{4}=-\frac{1}{12}

eq - 1\)

• \(m_3\times m_4=(-4)\times\frac{1}{4}=-1\)

Only one pair (\(y = - 4x-5\) and \(y=\frac{1}{4}x - 5\)) has slopes with a product of \(-1\). But for a rectangle, we need two pairs of parallel lines and four pairs of adjacent perpendicular lines.

Answer:

Shannon's error is that there are no parallel lines (since no two slopes are equal) and only one pair of perpendicular lines (whereas a rectangle requires two pairs of parallel lines and four pairs of adjacent perpendicular lines).