Question

in the diagram, \\( \overline { b d } \\) is a diameter of circle a. if the slope of \\( \overleftrightarrow { b e } \\) is \\( \frac { 3 } { 2 } \\), what is the slope of \\( \overleftrightarrow { b d } \\)?

Explanation:

Step1: Recall the property of perpendicular lines

If two lines are perpendicular, the product of their slopes is \(- 1\). Let the slope of \(\overleftrightarrow{BD}\) be \(m_1\) and the slope of \(\overleftrightarrow{BE}\) be \(m_2\). We know that \(m_2=\frac{3}{2}\).

Step2: Use the perpendicular - slope formula

Since \(\overleftrightarrow{BD}\perp\overleftrightarrow{BE}\) (a diameter is perpendicular to the tangent at the point of tangency), we use the formula \(m_1\times m_2=-1\). Substitute \(m_2 = \frac{3}{2}\) into the formula: \(m_1\times\frac{3}{2}=-1\).

Step3: Solve for \(m_1\)

To find \(m_1\), we can rewrite the equation \(m_1\times\frac{3}{2}=-1\) as \(m_1=-\frac{2}{3}\).

Answer:

\(-\frac{2}{3}\)