Straight-line explorer

<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: center; --cell-justify: center">Relationship</th><th style="text-align: center; --cell-justify: center">Condition</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">Parallel nonvertical lines</td><td style="text-align: center; --cell-justify: center">Equal gradients. Distinct parallel lines have different intercepts.</td></tr><tr><td style="text-align: center; --cell-justify: center">Perpendicular nonvertical lines</td><td style="text-align: center; --cell-justify: center">\(m_1m_2=-1\), so \(m_2=-\dfrac1{m_1}\).</td></tr><tr><td style="text-align: center; --cell-justify: center">Horizontal and vertical lines</td><td style="text-align: center; --cell-justify: center">These are perpendicular; use their directions instead of multiplying gradients.</td></tr></tbody></table></div></div>
<p>To find a related line, first determine its gradient, then use the given point to obtain its equation. Check both the point and the required gradient relationship.</p>
<p>For a quadrilateral, equal opposite-side gradients establish parallel sides. To prove a rectangle, establish a parallelogram and a right angle; one pair of perpendicular sides alone is insufficient.</p>
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<p>The lines \(y=\dfrac12x\) and \(y=-2x\) are perpendicular because&nbsp;\(\dfrac12(-2)=-1\).</p>

To find a related line, first determine its gradient, then use the given point to obtain its equation. Check both the point and the required gradient relationship.

For a quadrilateral, equal opposite-side gradients establish parallel sides. To prove a rectangle, establish a parallelogram and a right angle; one pair of perpendicular sides alone is insufficient.

The lines \(y=\dfrac12x\) and \(y=-2x\) are perpendicular because \(\dfrac12(-2)=-1\).

Parallel and perpendicular lines

Parallel and perpendicular lines

Relationship	Condition
Parallel nonvertical lines	Equal gradients. Distinct parallel lines have different intercepts.
Perpendicular nonvertical lines	\(m_1m_2=-1\), so \(m_2=-\dfrac1{m_1}\).
Horizontal and vertical lines	These are perpendicular; use their directions instead of multiplying gradients.