91Ó°ÊÓ

In mathematics, the coordinate plane is divided into four sections called quadrants. The first quadrant is situated in the top-right corner of this plane. Here, both the x and y coordinates are positive.
This is an important characteristic to remember, especially in geometry, because certain rules and lengths are easier to compute with positive values.
Kicking off with our exercise, when we talk about the right triangle being formed in the first quadrant, it tells us that the vertices lie in a region where both x and y are positive. This makes calculations straightforward since there are no negative distances to factor into our hypotenuse calculations.

The equation of a line provides a foundational model for describing straight paths on a coordinate plane. When expressed in the form of y = mx + b, it tells us two key things. The first, 'm,' is the slope of the line, indicating how steep the line is. The second, 'b,' is the y-intercept, showing where the line crosses the y-axis.
In our exercise, the line passing through the origin (0,0) and the point (3,2) can be expressed using the slope-intercept form. To find the slope 'm', simply compute rise over run, or the difference in y-values over the difference in x-values:

• Rise: Change in y-values (2 - 0)
• Run: Change in x-values (3 - 0)

This results in a slope (m) of 2/3. Hence, the equation of our line becomes y = (2/3)x. This equation helps us identify any corresponding y-value for a given x-value as the line traverses the first quadrant.

The Pythagorean Theorem is a crucial concept in geometry, especially when dealing with right triangles. It states that for any right triangle, the square of the hypotenuse length (longest side) is equal to the sum of the squares of the other two sides. Mathematically, this is written as \ \[ c^2 = a^2 + b^2 \] where 'c' represents the hypotenuse, and 'a' and 'b' are the triangle's other two sides.
In the context of our exercise, the base and height of the triangle are along the x-axis and y-axis respectively. We know that from the origin to any point \(x, y\), these can be essentially treated as the right triangle’s legs. Inserting these into the equation, we use our line equation y = \(\frac{2}{3}x\) to substitute for y, resulting in: \ \[ L = \sqrt{x^2 + (\frac{2x}{3})^2} = \frac{x\sqrt{13}}{3} \] This means the hypotenuse, 'L,' effectively becomes a function of 'x,' describing how its length would change as the x-value increases, adhering to the Pythagorean Theorem.