Square Translation And Point Location A Geometry Problem

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In the realm of geometry, understanding transformations and coordinate geometry is paramount. This article delves into a fascinating problem involving the translation of a square and the subsequent determination of a point's location relative to the translated square. We will meticulously analyze the given information, which includes the vertices of the translated square R'S'T'U' and the coordinates of point S. Our primary goal is to identify which point, among a set of options, lies on a side of the original square RSTU. This exploration will not only enhance our understanding of geometric transformations but also sharpen our problem-solving skills in coordinate geometry.

Consider a square RSTU that undergoes a translation, resulting in a new square R'S'T'U'. The vertices of the translated square are given as R'(-8, 1), S'(-4, 1), T'(-4, -3), and U'(-8, -3). Additionally, we know that point S in the original square has coordinates (3, -5). The central question we aim to address is: which point, from a provided set of options, lies on a side of the original square RSTU?

Before we can pinpoint the location of points relative to the original square, it's crucial to thoroughly understand the characteristics of the translated square R'S'T'U'. The coordinates of its vertices, R'(-8, 1), S'(-4, 1), T'(-4, -3), and U'(-8, -3), provide valuable insights into its dimensions and orientation.

First, let's determine the side length of the square. We can calculate the distance between two adjacent vertices, such as R' and S', or S' and T'. Using the distance formula, which is derived from the Pythagorean theorem, we have:

Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

Calculating the distance between R'(-8, 1) and S'(-4, 1):

Distance = √[(-4 - (-8))² + (1 - 1)²] = √[(4)² + (0)²] = √16 = 4

Similarly, the distance between S'(-4, 1) and T'(-4, -3):

Distance = √[(-4 - (-4))² + (-3 - 1)²] = √[(0)² + (-4)²] = √16 = 4

This confirms that the side length of the square R'S'T'U' is 4 units. This side length is a crucial piece of information, as it will help us determine the dimensions of the original square as well, since translation preserves side lengths.

Next, let's analyze the orientation of the square. We can observe that the sides R'S' and U'T' are horizontal lines (since the y-coordinates of their endpoints are the same), and the sides S'T' and R'U' are vertical lines (since the x-coordinates of their endpoints are the same). This indicates that the sides of the square are parallel to the coordinate axes, which simplifies our calculations.

The center of the square R'S'T'U' can be found by averaging the coordinates of opposite vertices, such as R' and T', or S' and U'. Let's calculate the center using R'(-8, 1) and T'(-4, -3):

Center x-coordinate = (-8 + (-4)) / 2 = -12 / 2 = -6 Center y-coordinate = (1 + (-3)) / 2 = -2 / 2 = -1

Therefore, the center of the translated square R'S'T'U' is (-6, -1). Understanding the center of the square is vital because it serves as a reference point for determining the translation vector.

The translation vector is the key to understanding how the original square RSTU was moved to its new position R'S'T'U'. To find this vector, we need to compare the coordinates of a point in the original square with its corresponding point in the translated square. We are given that point S has coordinates (3, -5) and its translated counterpart, S', has coordinates (-4, 1).

The translation vector can be calculated by subtracting the coordinates of the original point from the coordinates of the translated point:

Translation vector x-component = S'x - Sx = -4 - 3 = -7 Translation vector y-component = S'y - Sy = 1 - (-5) = 1 + 5 = 6

Therefore, the translation vector is (-7, 6). This means that every point in the original square was moved 7 units to the left and 6 units upwards to form the translated square. This translation vector is fundamental to finding the coordinates of the other vertices of the original square.

Now that we have the translation vector (-7, 6), we can determine the coordinates of the other vertices of the original square RSTU. We know the coordinates of S (3, -5). To find the coordinates of the other vertices, we need to reverse the translation process by adding the translation vector to the coordinates of the corresponding vertices in the translated square.

To find R, we add the translation vector to R'(-8, 1):

Rx = R'x + 7 = -8 + 7 = -1 Ry = R'y - 6 = 1 - 6 = -5

So, the coordinates of R are (-1, -5).

To find T, we add the translation vector to T'(-4, -3):

Tx = T'x + 7 = -4 + 7 = 3 Ty = T'y - 6 = -3 - 6 = -9

So, the coordinates of T are (3, -9).

To find U, we add the translation vector to U'(-8, -3):

Ux = U'x + 7 = -8 + 7 = -1 Uy = U'y - 6 = -3 - 6 = -9

So, the coordinates of U are (-1, -9).

Therefore, the vertices of the original square RSTU are R(-1, -5), S(3, -5), T(3, -9), and U(-1, -9). Knowing the coordinates of the original square's vertices is essential for determining which point lies on its side.

Now that we have the coordinates of the vertices of the original square RSTU, we can determine which point, from a given set of options, lies on a side of the square. The sides of the square are line segments connecting the vertices: RS, ST, TU, and UR.

To determine if a point lies on a side, we can use the equation of the line segment or check if the point lies within the boundaries defined by the vertices. Let's analyze the sides of the square:

  • Side RS: This is a horizontal line segment connecting R(-1, -5) and S(3, -5). The equation of this line is y = -5. Any point on this side will have a y-coordinate of -5 and an x-coordinate between -1 and 3, inclusive.
  • Side ST: This is a vertical line segment connecting S(3, -5) and T(3, -9). The equation of this line is x = 3. Any point on this side will have an x-coordinate of 3 and a y-coordinate between -5 and -9, inclusive.
  • Side TU: This is a horizontal line segment connecting T(3, -9) and U(-1, -9). The equation of this line is y = -9. Any point on this side will have a y-coordinate of -9 and an x-coordinate between -1 and 3, inclusive.
  • Side UR: This is a vertical line segment connecting U(-1, -9) and R(-1, -5). The equation of this line is x = -1. Any point on this side will have an x-coordinate of -1 and a y-coordinate between -5 and -9, inclusive.

To find which point lies on the side of the square RSTU, it would be necessary to have the options from the initial question. Let's consider a hypothetical point P(1, -5).

To determine if P lies on a side, we will check the following:

  • Does P lie on RS? The y-coordinate of P is -5, which matches the y-coordinate of the line RS. The x-coordinate of P is 1, which lies between -1 and 3. Therefore, P lies on the side RS.

If the point is not on the horizontal or vertical line then you need to use the point-slope form to verify if the point is on the line and use distance to check if the point is between the two vertices.

In summary, this exploration has taken us through a comprehensive analysis of a geometric translation problem. We began by dissecting the properties of the translated square R'S'T'U', calculated the crucial translation vector, and then successfully determined the coordinates of the original square's vertices. Understanding geometric transformations, coordinate geometry, and the properties of squares are fundamental in solving such problems. The key steps included:

  1. Analyzing the translated square to determine its dimensions and orientation.
  2. Calculating the translation vector by comparing corresponding points in the original and translated squares.
  3. Finding the vertices of the original square by reversing the translation.
  4. Using the coordinates of the vertices to define the equations of the sides of the square.

Finally, we discussed how to determine whether a given point lies on a side of the original square. This process demonstrates the power of combining geometric principles with algebraic techniques to solve intricate problems. This understanding not only enhances our mathematical prowess but also provides a solid foundation for tackling more complex geometric challenges in the future.