Rotational Transformations In Geometry Explained

Hey guys! Let's dive into a cool geometry problem where we explore how rotations can move points in a coordinate plane. We've got a polygon vertex initially at $(3, -2)$ that ends up at $(2, 3)$ after a transformation. Our mission? To figure out which rotations could have caused this move. Buckle up, because we're about to unravel some rotational mysteries!

The Problem at Hand

So, here’s the deal: imagine you have a shape, and one of its corners (a vertex) is chilling at the point $(3, -2)$. Then, poof, after some magical rotation, that same vertex is now hanging out at $(2, 3)$. The question we need to crack is, what kind of rotations could have made this happen? We need to consider both clockwise and counterclockwise rotations because, you know, geometry is all about options!

Visualizing the Transformation

Before we get into the nitty-gritty, let's take a moment to visualize what’s going on. Think of a standard coordinate plane, the kind you've probably doodled on a million times. Plot the points $(3, -2)$ and $(2, 3)$. You'll notice they're in different quadrants – $(3, -2)$ is in the fourth quadrant (where x is positive and y is negative), and $(2, 3)$ is in the first quadrant (where both x and y are positive). This change in quadrants is a big clue that a rotation is definitely involved. To really nail this, you might want to sketch it out on paper or use a graphing tool. Seeing the points in relation to each other can make the possible rotations much clearer.

Key Concepts: Rotations in the Coordinate Plane

Now, let's brush up on some essential concepts about rotations. A rotation is a transformation that turns a figure around a fixed point, which we usually call the center of rotation. In most problems, the center of rotation is the origin (0, 0), and that’s what we'll assume here. When we talk about rotations, we usually measure them in degrees, and we always specify whether the rotation is clockwise (like the hands on a clock) or counterclockwise (the opposite direction).

A crucial thing to remember is how rotations affect the coordinates of a point. A 90-degree counterclockwise rotation swaps the x and y coordinates and negates the new x-coordinate. A 180-degree rotation negates both coordinates. A 270-degree counterclockwise rotation (which is the same as a 90-degree clockwise rotation) swaps the x and y coordinates and negates the new y-coordinate. These rules are super handy for solving rotation problems, so keep them in your mental toolkit!

Distance from the Origin

Another vital aspect of rotations is that they preserve the distance from the center of rotation. In simpler terms, if our vertex is a certain distance from the origin before the rotation, it will be the same distance from the origin after the rotation. This is because a rotation is a rigid transformation – it doesn't stretch or squash the shape, it just turns it. So, let’s calculate the distance of both points from the origin using the distance formula, which is derived from the Pythagorean theorem: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. For our points $(3, -2)$ and $(2, 3)$, the distance from the origin $(0, 0)$ is:

For $(3, -2)$: $d = \sqrt{(3 - 0)^2 + (-2 - 0)^2} = \sqrt{3^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13}$

For $(2, 3)$: $d = \sqrt{(2 - 0)^2 + (3 - 0)^2} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}$

Notice anything? Both points are the same distance, $\sqrt{13}$, from the origin. This confirms that a rotation is indeed a plausible transformation, as the distance is preserved.

Possible Rotations

Okay, so we know a rotation is in play. But which rotations could have taken our vertex from $(3, -2)$ to $(2, 3)$? This is where things get interesting. We need to think about the angles of rotation, both clockwise and counterclockwise, that would achieve this transformation.

Visualizing the Angles

Imagine drawing lines from the origin to both points, $(3, -2)$ and $(2, 3)$. The angle between these lines represents the rotation we're trying to find. Now, this isn't a straightforward 90-degree or 180-degree rotation, so we'll need to dig a little deeper. We can use trigonometric functions to help us figure out the angles. Remember SOH CAH TOA? It's about to become your new best friend!

Using Trigonometry to Find the Angles

We can use the arctangent function (also known as the inverse tangent, or tan⁻¹) to find the angles. The arctangent gives us the angle whose tangent is a given value. Specifically, we'll use arctan(y/x) to find the angle each point makes with the positive x-axis.

For $(3, -2)$, the angle θ₁ is arctan(-2/3). This will give us a negative angle, since $(3, -2)$ is in the fourth quadrant. Using a calculator, we find that θ₁ ≈ -33.69 degrees. This is the angle measured clockwise from the positive x-axis.

For $(2, 3)$, the angle θ₂ is arctan(3/2). This point is in the first quadrant, so the angle will be positive. Using a calculator, we find that θ₂ ≈ 56.31 degrees.

Now, to find the rotation angle, we need to find the difference between these two angles. Since θ₁ is negative (clockwise) and θ₂ is positive (counterclockwise), we'll add their absolute values to find the total rotation angle.

Calculating the Rotation Angles

To find the counterclockwise rotation angle, we add the absolute value of θ₁ to θ₂: Counterclockwise Rotation = |θ₁| + θ₂ ≈ |-33.69| + 56.31 ≈ 33.69 + 56.31 ≈ 90 degrees.

This tells us that a 90-degree counterclockwise rotation could be one of the transformations!

But hold on, there’s more! We also need to consider clockwise rotations. To find the clockwise rotation angle, we can subtract the counterclockwise rotation from 360 degrees: Clockwise Rotation = 360 - 90 = 270 degrees.

So, a 270-degree clockwise rotation could also do the trick.

Selecting the Correct Options

Now that we've done the math and the visualization, we can confidently say that the possible transformations are a 90-degree counterclockwise rotation and a 270-degree clockwise rotation. These rotations would indeed move the vertex from $(3, -2)$ to $(2, 3)$.

Conclusion

Geometry can be like a puzzle, but with the right tools and concepts, we can solve even the trickiest problems. By understanding rotations, using trigonometry, and visualizing transformations, we cracked this problem like pros. Keep practicing, and you'll become a geometry whiz in no time! Remember, guys, math is all about the journey, not just the destination. So, enjoy the ride and keep those brains spinning!