Quadrilateral Translation: Find New Coordinates Easily

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Find the Coordinates of Quadrilateral P Q R S with Translation (x, y) → (x-5, y+3)

Alright guys, let's dive into some coordinate geometry! We're going to take a quadrilateral, move it around a bit using translation, and then find the new coordinates. Sounds like fun, right? Our quadrilateral is named PQRS, and we've got the coordinates of its vertices: P(1,4), Q(-1,4), R(-2,-4), and S(2,-4). The translation we're applying is (x, y) → (x-5, y+3). This means we're shifting every point 5 units to the left and 3 units up. Let's break it down step by step.

Understanding Translation in Coordinate Geometry

Before we jump into the calculations, let's quickly recap what translation means in coordinate geometry. Translation is essentially moving every point of a shape the same distance in the same direction. Think of it as sliding the shape across the plane without rotating or resizing it. The rule (x, y) → (x-5, y+3) tells us exactly how each point will be moved. The x-coordinate decreases by 5, and the y-coordinate increases by 3. This type of transformation is fundamental in various fields, including computer graphics, game development, and even robotics, where understanding spatial relationships and movements is crucial.

In practical terms, if you have a point (x, y), applying this translation means you subtract 5 from its x-coordinate and add 3 to its y-coordinate to find its new location (x', y'). So, x' = x - 5 and y' = y + 3. This simple yet powerful concept allows us to manipulate shapes and objects in a predictable manner. Understanding translations helps in visualizing how objects move and interact in space, which is super useful in fields that deal with spatial data and transformations.

Moreover, grasping the concept of translation sets the stage for understanding more complex transformations like rotations, reflections, and scaling. Each of these transformations alters the position or size of a shape according to specific rules, and mastering the basics of translation provides a solid foundation for tackling these advanced topics. So, when you encounter a translation problem, remember that it's all about shifting each point by a consistent amount in a consistent direction.

Applying the Translation to Each Vertex

Now, let's apply this translation to each vertex of our quadrilateral. We'll go through each point one by one to find the new coordinates after the translation.

Vertex P(1,4)

For vertex P(1,4), we apply the translation (x, y) → (x-5, y+3). So, the new x-coordinate will be 1 - 5 = -4, and the new y-coordinate will be 4 + 3 = 7. Therefore, the new coordinates of P, which we'll call P', are (-4, 7). This means P has moved 5 units to the left and 3 units up.

Vertex Q(-1,4)

Next up is vertex Q(-1,4). Applying the same translation, the new x-coordinate will be -1 - 5 = -6, and the new y-coordinate will be 4 + 3 = 7. Thus, the new coordinates of Q, which we'll call Q', are (-6, 7). Notice how Q also moves 5 units to the left and 3 units up, maintaining the same vertical position as P after the translation.

Vertex R(-2,-4)

Now let's transform vertex R(-2,-4). Applying the translation (x, y) → (x-5, y+3), the new x-coordinate will be -2 - 5 = -7, and the new y-coordinate will be -4 + 3 = -1. Hence, the new coordinates of R, which we'll call R', are (-7, -1). This point has shifted significantly, moving both left and slightly upwards.

Vertex S(2,-4)

Finally, we transform vertex S(2,-4). Applying the translation, the new x-coordinate will be 2 - 5 = -3, and the new y-coordinate will be -4 + 3 = -1. So, the new coordinates of S, which we'll call S', are (-3, -1). Like R, S has also moved left and upwards.

Summarizing the New Coordinates

Alright, let's put it all together. After applying the translation (x, y) → (x-5, y+3) to the vertices of quadrilateral PQRS, we found the new coordinates:

  • P'(-4, 7)
  • Q'(-6, 7)
  • R'(-7, -1)
  • S'(-3, -1)

So, the correct answer is A. P'(-4,7), Q'(-6,7), R'(-7,-1) and S'(-3,-1). This is how we find the coordinates of a translated quadrilateral. Remember, the key is to apply the translation rule consistently to each vertex.

Visualizing the Translation

To really understand what's happening, it helps to visualize the translation. Imagine plotting the original quadrilateral PQRS on a coordinate plane. Then, for each point, shift it 5 units to the left and 3 units up. You'll see the new quadrilateral P'Q'R'S' in its new location. Visualizing these transformations can make coordinate geometry much more intuitive.

Consider using graphing software or even a simple sketch on paper to plot the points. Start by drawing the x and y axes and plotting the original points P(1,4), Q(-1,4), R(-2,-4), and S(2,-4). Connect these points to form the quadrilateral PQRS. Next, apply the translation to each point and plot the new points P'(-4,7), Q'(-6,7), R'(-7,-1), and S'(-3,-1). Connect these new points to form the translated quadrilateral P'Q'R'S'.

By comparing the positions of the original and translated quadrilaterals, you can clearly see the effect of the translation. The entire shape has shifted 5 units to the left and 3 units up, but its size and shape remain unchanged. This visual confirmation reinforces the concept of translation as a movement of the entire shape without any distortion.

Practical Applications of Coordinate Translation

Now, you might be wondering,