91Ó°ÊÓ

Plotting points on a graph is an essential skill in coordinate geometry. It's like marking spots on a map. To plot a point, you need an ordered pair \((x, y)\) which tells you where to place the dot on a grid. The first number, \(x\), is the horizontal position (left or right), while the second number, \(y\), is the vertical position (up or down). For example, let’s take the points \((6, -3)\) and \((6, 5)\). Both have the same \(x\)-coordinate, which means they sit on the same vertical line at \(x = 6\). You simply move up or down to reach the correct \(y\)-value for each point. Plotting these tells us a lot quickly. Since the \(x\)-coordinates match, you already know they form a vertical line.

Calculating the distance between two points on a graph helps determine how far apart they are. This can be visualized as the length of a straight line connecting them. We use the distance formula:\[ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]This formula comes from the Pythagorean theorem, combining the change in \(x\) and \(y\). Here’s how it works with points \((6, -3)\) and \((6, 5)\):

• Both \(x_1\) and \(x_2\) are 6, so \((x_2 - x_1)^2\) is zero.
• The change in \(y\) is \(5 - (-3) = 8\).
• The distance becomes \(\sqrt{0 + 8^2 } = 8\).

So, the distance between these two points is 8 units, a reflection of their separation along the vertical line.

The midpoint of a line segment gives the central point between two ends. It’s like finding the middle point on a ruler to have equal sides. We use:\[ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]This formula averages the \(x\) and \(y\) coordinates, helping locate the midpoint accurately. With points \((6, -3)\) and \((6, 5)\):

• \(x_1\) and \(x_2\) are both 6, so \(\frac{6 + 6}{2} = 6\).
• For \(y\), calculate \(\frac{-3 + 5}{2}\) which is 1.

This gives a midpoint of \((6, 1)\). It’s directly between the two original points on the vertical path, showing balance and symmetry.