Question

Short Response Given \(L(2,1), M(4,3), P(0,2)\), and \(Q(3,-1)\), determine if \(\overline{L M}\) and \(\overline{P Q}\) are parallel, perpendicular, or neither. (Lesson 4-5)

Short answer

The lines \(\overline{LM}\) and \(\overline{PQ}\) are perpendicular.

Step-by-step solution

Find Slope of Line LM

The slope of a line through two points \(x_1, y_1\) and \(x_2, y_2\) is given by the formula \\[ m = \frac{y_2 - y_1}{x_2 - x_1} \\] Applying this to points \(L(2,1)\) and \(M(4,3)\), the slope is \\[ m_{LM} = \frac{3 - 1}{4 - 2} = \frac{2}{2} = 1 \\] So, the slope of \(\overline{LM}\) is 1.

Find Slope of Line PQ

Using the same formula for slope, apply it to points \(P(0,2)\) and \(Q(3,-1)\): \\[ m_{PQ} = \frac{-1 - 2}{3 - 0} = \frac{-3}{3} = -1 \\] Thus, the slope of \(\overline{PQ}\) is -1.

Determine Relationship of Slopes

To determine the relationship of the lines, check their slopes. If the slopes are equal, the lines are parallel. If the product of the slopes is \(-1\), the lines are perpendicular:- Slope of \(\overline{LM} = 1\)- Slope of \(\overline{PQ} = -1\)- Check the product: \(1 \times -1 = -1\)Since their product is \(-1\), the lines are perpendicular.

Key concepts

slope of a line

The slope of a line is an essential concept in geometry that defines how steep a line is. It helps us understand the direction and angle of the line. Mathematically, the slope is expressed as the ratio of the vertical change to the horizontal change between two points on the line.
For any two given points - \(x_1, y_1\)- \(x_2, y_2\)The formula for slope is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]The symbol \(m\) denotes the slope. A positive slope indicates the line rises moving from left to right, while a negative slope shows it falls. A zero slope means a horizontal line, and an undefined slope signifies a vertical line.

parallel lines

Parallel lines are lines in a plane that remain the same distance apart and never meet, no matter how far they are extended. In the context of slopes, parallel lines will always have the same slope.
This makes it easy to determine if two lines are parallel: simply compare their slopes.Here's the rule of thumb:- If \(m_1 = m_2\), then the lines are parallel.It means that the changes in the y-coordinates relative to the changes in the x-coordinates are consistently proportionate.

perpendicular lines

Perpendicular lines are lines that intersect to form right angles, or 90-degree angles. When we relate this concept to slopes, two lines are perpendicular if the product of their slopes is \(-1\).
This is because the slopes of perpendicular lines are negative reciprocals of each other.For example, if one line has a slope \(m_1 = a\), then a perpendicular line's slope would be \(m_2 = -\frac{1}{a}\) making the product:- \(m_1 \times m_2 = a \times -\frac{1}{a} = -1\)This rule helps to quickly determine whether two lines are perpendicular.

slope formula

The slope formula is vital in calculating the steepness or angle of a line between two given points. As stated earlier, the formula is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This formula is straightforward to use:

• Identify two clear points on the line.
• Subtract the y-coordinate of the first point from the y-coordinate of the second.
• Subtract the x-coordinate of the first point from the x-coordinate of the second.
• Divide the difference in y values by the difference in x values to get the slope.

This slope formula not only reveals whether lines are parallel or perpendicular but also provides insights into their direction and orientation.