Finding The Equation Of A Line Through The Origin And Given Points

Hey there, math enthusiasts! Today, we're diving into an exciting problem that combines geometry and algebra. We're going to figure out the equation of a line that passes through the origin and two other points, $(6, r)$ and $(2r, 9)$, where $r$ is a positive constant. This is a classic problem that tests our understanding of linear equations and how to work with coordinates. So, buckle up and let's get started!

The Problem at Hand

The problem states that we have a line that gracefully glides through three specific points the origin (0, 0), (6, r), and (2r, 9). The twist here is that 'r' is a positive constant, meaning it's a number greater than zero. Our mission, should we choose to accept it, is to find the equation that perfectly describes this line. We're given a few options, and only one will fit the bill:

A) $y=\frac{\sqrt{3}}{2} x$ B) $y=\sqrt{3} x$ C) $y=2 \sqrt{3} x$ D) $y=3x$

Let's break down how we can crack this problem.

Finding the Slope: The Key to Unlocking the Equation

Understanding the Concept of Slope

The slope of a line is a fundamental concept in coordinate geometry. It tells us how steep the line is and in what direction it's inclined. Mathematically, the slope (often denoted by 'm') is defined as the change in the y-coordinate divided by the change in the x-coordinate between any two points on the line. Think of it as "rise over run." A positive slope indicates that the line is going upwards as you move from left to right, while a negative slope means it's going downwards.

Calculating the Slope Using Two Points

The formula to calculate the slope (m) given two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

This formula is our trusty tool for finding the slope of the line in this problem.

Applying the Slope Formula to Our Points

Since our line passes through the origin (0, 0), we can use this as one of our points. Let's use the point (6, r) along with the origin. Plugging these values into the slope formula, we get:

$m = \frac{r - 0}{6 - 0} = \frac{r}{6}$

So, the slope of our line is $\frac{r}{6}$.

Now, let's use the origin (0, 0) and the point (2r, 9) to calculate the slope again:

$m = \frac{9 - 0}{2r - 0} = \frac{9}{2r}$

We now have two expressions for the slope: $\frac{r}{6}$ and $\frac{9}{2r}$. Since these both represent the slope of the same line, they must be equal. This gives us a crucial equation to solve for 'r'.

Solving for 'r': Unraveling the Constant

Setting the Slopes Equal

We've established that $\frac{r}{6} = \frac{9}{2r}$. This equation is the key to finding the value of 'r'. To solve for 'r', we'll use a technique called cross-multiplication.

Cross-Multiplication

Cross-multiplication is a handy way to solve equations involving fractions. It works like this: if you have an equation of the form $\frac{a}{b} = \frac{c}{d}$, then you can cross-multiply to get $ad = bc$. In our case, this means:

$r * 2r = 6 * 9$

Simplifying the Equation

Let's simplify the equation:

$2r^2 = 54$

To isolate $r^2$, we divide both sides by 2:

$r^2 = 27$

Finding the Value of 'r'

Now, we need to find the square root of 27 to solve for 'r'. Remember that 'r' is a positive constant, so we only consider the positive square root.

$r = \sqrt{27}$

We can simplify $\sqrt{27}$ by factoring out the largest perfect square, which is 9:

$r = \sqrt{9 * 3} = \sqrt{9} * \sqrt{3} = 3\sqrt{3}$

So, we've found that $r = 3\sqrt{3}$. This is a significant step forward!

Determining the Slope: Plugging 'r' Back In

Using the Value of 'r'

Now that we know $r = 3\sqrt{3}$, we can plug this value back into either of our slope expressions. Let's use the simpler one, $m = \frac{r}{6}$:

$m = \frac{3\sqrt{3}}{6}$

Simplifying the Slope

We can simplify this fraction by dividing both the numerator and denominator by 3:

$m = \frac{\sqrt{3}}{2}$

Fantastic! We've found the slope of our line: $m = \frac{\sqrt{3}}{2}$.

Finding the Equation: The Grand Finale

The Slope-Intercept Form

The equation of a line in slope-intercept form is given by:

$y = mx + b$

where 'm' is the slope and 'b' is the y-intercept. Since our line passes through the origin (0, 0), the y-intercept 'b' is 0. This simplifies our equation to:

$y = mx$

Plugging in the Slope

We know the slope is $m = \frac{\sqrt{3}}{2}$. Plugging this into our equation, we get:

$y = \frac{\sqrt{3}}{2}x$

The Answer!

This equation matches option A! So, the equation that describes the graph of the line is $y = \frac{\sqrt{3}}{2}x$.

Wrapping Up: A Triumph of Math!

Recap of Our Journey

We started with a problem that seemed a bit complex, but by breaking it down step-by-step, we were able to solve it. Here's a quick recap of our journey:

1. We understood the problem and identified the key information: a line passing through the origin, (6, r), and (2r, 9).
2. We used the slope formula to find two expressions for the slope: $\frac{r}{6}$ and $\frac{9}{2r}$.
3. We set these expressions equal to each other and solved for 'r', finding that $r = 3\sqrt{3}$.
4. We plugged the value of 'r' back into our slope expression to find the slope: $m = \frac{\sqrt{3}}{2}$.
5. We used the slope-intercept form of a line ($y = mx + b$) and the fact that the line passes through the origin (b = 0) to find the equation: $y = \frac{\sqrt{3}}{2}x$.

Key Takeaways

This problem highlights the power of using fundamental concepts like slope and the slope-intercept form to solve geometric problems. It also shows how setting up and solving equations is a crucial skill in mathematics. Keep practicing, and you'll become a master problem-solver!

So, there you have it, guys! We successfully navigated this problem and found the equation of the line. Keep your mathematical minds sharp, and until next time, happy problem-solving!