91Ó°ÊÓ

The distance formula is a tool used in coordinate geometry to determine the length between two points. Think of it as a way to extend the Pythagorean theorem into the grid of a plane. You can visualize a line segment as the hypotenuse of a right triangle, with the horizontal and vertical sides determined by the differences in the x-coordinates and y-coordinates of the points. Here's the formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] To use it, simply plug the x and y values of your two points into the formula. For example, with points \( P(3, -8) \) and \( Q(-5, 2) \), subtract the x-values: \( -5 - 3 = -8 \). For the y-values: \( 2 - (-8) = 10 \). Plug these into the formula: - \( d = \sqrt{(-8)^2 + (10)^2} \)- \( d = \sqrt{64 + 100} \)- \( d = \sqrt{164} \) This gives the distance of the line segment \( \overline{PQ} \) in the coordinate plane.

Slope is a way of describing the steepness and direction of a line. It's often represented by the letter \( m \) and is calculated by dividing the change in the y-coordinates by the change in the x-coordinates between two points. The formula is simple: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] In practical terms, this means: - If \( m \) is positive, the line rises from left to right.- If \( m \) is negative, the line declines from left to right.- If \( m \) is zero, the line is perfectly horizontal.- If the denominator is zero, the line is perfectly vertical, and the slope is undefined. Using the points \( P(3, -8) \) and \( Q(-5, 2) \), the slope is found by: - \( m = \frac{2 - (-8)}{-5 - 3} = \frac{10}{-8} = -\frac{5}{4} \) This negative slope indicates our line segment is indeed descending from left to right.

The midpoint formula is used to find the exact middle point of a line segment in a two-dimensional space. This is incredibly useful when you need to find a point that partitions a segment into two equal halves. The formula is: \[ (x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] With points \( P(3, -8) \) and \( Q(-5, 2) \), you compute:- The x-coordinate of the midpoint: \( \frac{3 + (-5)}{2} = -1 \)- The y-coordinate of the midpoint: \( \frac{-8 + 2}{2} = -3 \) Hence, the midpoint \( M \) of \( \overline{PQ} \) is \( (-1, -3) \). This point divides the line segment into two equal parts while accurately representing the average of the x-coordinates and y-coordinates of the endpoints.