Translating Points Finding The Image Of (9, -5)

In the realm of coordinate geometry, transformations play a crucial role in altering the position and orientation of geometric figures. One fundamental transformation is translation, which involves shifting a figure without changing its size or shape. In this article, we will delve into the concept of translation and apply it to a specific point, $(9, -5)$, under the transformation rule $(x, y) ightarrow (x + 7, y - 2)$. Our goal is to determine the resulting image of the point after the translation.

Understanding Translations in Coordinate Geometry

In coordinate geometry, a translation is a transformation that shifts every point of a figure by the same distance in a given direction. This direction is defined by a translation vector, which specifies the horizontal and vertical components of the shift. The transformation rule $(x, y) ightarrow (x + a, y + b)$ represents a translation where every point $(x, y)$ is shifted $a$ units horizontally and $b$ units vertically. The values of $a$ and $b$ determine the direction and magnitude of the translation.

To visualize this, imagine a point on a coordinate plane. Translating this point means sliding it along the plane without rotating or reflecting it. The translation vector acts as a guide, indicating how far and in what direction the point should move. For instance, if the translation vector is $(3, 2)$, the point will move 3 units to the right and 2 units upwards.

Translations preserve the shape and size of the figure being transformed. This means that if you translate a line segment, the resulting line segment will have the same length and orientation. Similarly, if you translate a polygon, the resulting polygon will be congruent to the original one. This property makes translations fundamental in various geometric constructions and proofs.

Understanding translations is essential for various applications, including computer graphics, robotics, and physics. In computer graphics, translations are used to move objects on the screen. In robotics, robots use translations to navigate and manipulate objects. In physics, translations describe the motion of objects in space.

Applying the Translation Rule (x, y) → (x + 7, y - 2) to the Point (9, -5)

Now, let's apply the given translation rule, $(x, y) ightarrow (x + 7, y - 2)$, to the point $(9, -5)$. This rule indicates that we need to shift the point 7 units horizontally and -2 units vertically. In other words, we add 7 to the x-coordinate and subtract 2 from the y-coordinate.

To find the image of the point $(9, -5)$ under this translation, we perform the following calculations:

• New x-coordinate: $x' = x + 7 = 9 + 7 = 16$
• New y-coordinate: $y' = y - 2 = -5 - 2 = -7$

Therefore, the image of the point $(9, -5)$ after the translation is $(16, -7)$.

This process demonstrates how a translation rule transforms a point in the coordinate plane. By adding the horizontal component of the translation vector to the x-coordinate and the vertical component to the y-coordinate, we obtain the new coordinates of the translated point. The resulting point, $(16, -7)$, represents the new position of the original point after the shift.

Visualizing this translation on a coordinate plane can further enhance understanding. Imagine plotting the point $(9, -5)$ and then moving it 7 units to the right and 2 units down. The new position of the point will be $(16, -7)$, confirming our calculations.

Analyzing the Result and its Significance

The result of translating the point $(9, -5)$ using the rule $(x, y) ightarrow (x + 7, y - 2)$ is the point $(16, -7)$. This means that the original point has been shifted 7 units to the right and 2 units downwards on the coordinate plane. The transformation rule effectively alters the position of the point while preserving its inherent properties.

This specific translation is an example of a rigid transformation, which means that it preserves the distance between points. The distance between any two points in the original figure will be the same as the distance between their corresponding images after the translation. This property is crucial in various geometric applications, as it ensures that the shape and size of the figure remain unchanged during the transformation.

The significance of this result lies in its ability to demonstrate the fundamental principles of translations in coordinate geometry. By applying a translation rule to a point, we can predict its new location after the transformation. This understanding is essential for various mathematical and real-world applications, including:

• Computer graphics: Translations are used to move objects on the screen, create animations, and design user interfaces.
• Robotics: Robots use translations to navigate their environment, manipulate objects, and perform tasks in a precise manner.
• Physics: Translations describe the motion of objects in space, such as the movement of a projectile or the trajectory of a satellite.
• Mapping and navigation: Translations are used to create maps, plan routes, and determine the location of objects.

Conclusion: The Image of (9, -5) Under the Transformation is (16, -7)

In conclusion, by applying the translation rule $(x, y) ightarrow (x + 7, y - 2)$ to the point $(9, -5)$, we have determined that the resulting image is $(16, -7)$. This transformation shifts the point 7 units horizontally and -2 units vertically, effectively changing its position on the coordinate plane while preserving its inherent properties. The concept of translation is fundamental in coordinate geometry and has numerous applications in various fields, including computer graphics, robotics, physics, and mapping. Understanding translations allows us to manipulate geometric figures, predict their new positions, and solve various problems related to spatial transformations.

Therefore, the correct answer to the question "What's the result of translating the point $(9,-5)$ using the rule $(x, y) ightarrow(x+7, y-2)$?" is A) $(16, -7)$.