91Ó°ÊÓ

Coordinate geometry, or analytic geometry, is the study of geometry using a coordinate system. It helps us evaluate the position of points using ordered pairs, such as \((x, y)\) in a two-dimensional plane. In this exercise, points are defined using coordinates on an x-y plane.

The concept of finding distances and midpoints between two points is fundamental. For example, when we look at the length between two points on a horizontal line segment, like from \(C(2, -5)\) to \(D(10, -5)\), we only need to find the difference in the x-coordinates, because the y-coordinates remain constant. This is an application of simple arithmetic in coordinate geometry.

In addition, the midpoint between two points with known coordinates can easily be found using the formula:

• Midpoint, \(M\), between points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]This allows us to easily find the x-coordinate of the point \(P\) in the given exercise.

One unique property of a 30-60-90 triangle is its specific side length ratios. These triangles have sides in a fixed ratio of \(1: \sqrt{3}: 2\). This means:

• The side opposite the \(30^\circ\) angle, known as the short leg, is the shortest side, and serves as the base measure.
• The hypotenuse, or the side opposite the \(90^\circ\) angle, is twice the length of the short leg.
• The side opposite the \(60^\circ\) angle, or the long leg, is \(\sqrt{3}\) times the length of the short leg.

In this problem, given that the hypotenuse \(CD\) is 8, we can easily determine:

• Short leg \(PC\) is half of \(CD\), which equals 4.
• Long leg \(PD\) is \(4\sqrt{3}\) because it is \(\sqrt{3}\) times the short leg.

Understanding these ratios allows for straightforward calculation without needing complex trigonometry.

The hypotenuse in a right triangle is the side opposite the right angle. It's the longest side and plays a key role when working with triangles in both coordinate geometry and standard plane geometry.

To calculate the hypotenuse when you know two vertices of the triangle, like \(C(2, -5)\) and \(D(10, -5)\), you take the distance between these two points. For points aligned horizontally like in this problem, the hypotenuse length is merely the difference in x-values:

• \(CD = 10 - 2 = 8\)

This is crucial for setting the lengths for other sides based on known triangle ratios like in the 30-60-90 triangle. From there, the hypotenuse helps in further identifying properties of the triangle, confirming that the solution is correctly derived.