Given points $a$ and $b$ on the coordinate plane, which choice of coordinates for points $a^{\text{'}}$ and $b^{\text{'}}$ would help prove that lines $ab$ and $a^{\text{'}}{b}^{\text{'}}$ are perpendicular?

• A. Choose $a^{\text{'}}$ and $b^{\text{'}}$ so that both lines pass through the origin
• B. Choose $a^{\text{'}}$ and $b^{\text{'}}$ so that the product of the slopes of $ab$ and $a^{\text{'}}{b}^{\text{'}}$ is $-1$
• C. Choose $a^{\text{'}}$ and $b^{\text{'}}$ so that the lines are parallel
• D. Choose $a^{\text{'}}$ and $b^{\text{'}}$ so that the slopes of $ab$ and $a^{\text{'}}{b}^{\text{'}}$ are equal

Answer: (B)

Verified step by step guidance

1. Recall that two lines are perpendicular if and only if the product of their slopes is -1.
2. Find the slope of the line passing through points a and b using the formula \(\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}\).
3. Choose coordinates for points \(a'\) and \(b'\) such that the slope of the line \(a'b'\) satisfies the condition that when multiplied by the slope of line \(ab\), the product is -1.
4. This means if the slope of \(ab\) is \(m\), then the slope of \(a'b'\) should be \(-\frac{1}{m}\) to ensure perpendicularity.
5. By selecting \(a'\) and \(b'\) to satisfy this slope condition, you can prove that lines \(ab\) and \(a'b'\) are perpendicular.