Line Segment Dilation Problem Calculation Of Length A'A

In the realm of geometry, dilation is a transformation that alters the size of a figure without changing its shape. This transformation is defined by a center of dilation and a scale factor. The center of dilation is a fixed point around which the figure is either enlarged or reduced, while the scale factor determines the extent of this enlargement or reduction. A scale factor between 0 and 1 results in a reduction (also known as a contraction or shrinking), whereas a scale factor greater than 1 results in an enlargement. When working with dilations, it's crucial to understand how distances from the center of dilation to points on the original figure relate to distances from the center of dilation to corresponding points on the dilated figure. This relationship is directly governed by the scale factor. If we have a line segment AB that is dilated with center Q and scale factor k to produce line segment A'B', then the distance QA' will be k times the distance QA, and similarly, QB' will be k times the distance QB. This property is fundamental in solving problems involving dilations, as it allows us to set up proportions and equations to find unknown lengths. For instance, if we know the scale factor and the length of QA', we can determine the original length QA by dividing QA' by the scale factor. Conversely, if we know QA and the scale factor, we can find QA' by multiplying QA by the scale factor. These calculations are essential in various geometric applications, including mapmaking, scaling architectural designs, and computer graphics, where accurately resizing objects while maintaining their proportions is paramount. Understanding these principles of dilation enables us to solve a wide range of problems involving geometric transformations and measurements.

In this specific problem, we are given a line segment AB that has been dilated to create a new line segment A'B' using the dilation rule D Q, 0.15. This notation tells us that the center of dilation is point Q, and the scale factor is 0.15. Since the scale factor is less than 1, we know that the dilation is a reduction, meaning that the resulting line segment A'B' will be smaller than the original line segment AB. The problem states that the length of QA' is 2.4 units. This is the distance from the center of dilation Q to the point A' on the dilated line segment. We are also given that the length of A'A is x, which represents the distance between point A' on the dilated segment and point A on the original segment. Our goal is to find the value of x, which will tell us how much the point A has moved closer to the center of dilation as a result of the dilation. To solve this problem, we need to use the properties of dilation and the given information to set up an equation that relates the lengths QA, QA', and the scale factor. The key principle here is that the distance from the center of dilation to a point on the dilated figure is equal to the scale factor times the distance from the center of dilation to the corresponding point on the original figure. In other words, QA' = 0.15 * QA. This relationship is crucial for establishing the connection between the known length QA' and the unknown length QA. Once we find QA, we can use the fact that QA is the sum of QA' and A'A (since A' lies on the segment QA) to solve for x. This problem exemplifies how understanding the fundamental properties of geometric transformations like dilation can help us find unknown lengths and distances in geometric figures. By applying these principles, we can systematically approach and solve problems involving scaling and resizing of shapes.

To find the value of x, we will use the properties of dilation and the information provided in the problem. We know that the line segment AB was dilated to create line segment A'B' using a scale factor of 0.15 with point Q as the center of dilation. This means that the distance from the center of dilation to any point on the dilated segment is 0.15 times the distance from the center of dilation to the corresponding point on the original segment. Mathematically, this can be expressed as:

QA' = 0.15 * QA

We are given that the length of QA' is 2.4 units. So, we can substitute this value into the equation:

2. 4 = 0.15 * QA

Now, we need to solve for QA. To do this, we divide both sides of the equation by 0.15:

QA = 2.4 / 0.15

QA = 16 units

So, the length of QA is 16 units. Now we know the distance from the center of dilation Q to the original point A. The problem also tells us that the length of A'A is x. Since A' lies on the segment QA, we can write the relationship:

QA = QA' + A'A

We know QA = 16 units, QA' = 2.4 units, and A'A = x. Substituting these values into the equation, we get:

16 = 2.4 + x

Now, we solve for x by subtracting 2.4 from both sides of the equation:

x = 16 - 2.4

x = 13.6 units

Therefore, the length of A'A, which is x, is 13.6 units. This means that the distance between the dilated point A' and the original point A is 13.6 units. This result aligns with the understanding that since the scale factor is less than 1, the dilated segment A'B' is smaller and closer to the center of dilation Q compared to the original segment AB. This step-by-step solution demonstrates how the principles of dilation, combined with basic algebraic techniques, can be used to determine unknown lengths in geometric figures. Understanding these methods is crucial for tackling more complex problems in geometry and related fields.

In conclusion, by applying the principles of dilation and using the given information, we have successfully determined the value of x, which represents the length of A'A. The problem presented us with a scenario where line segment AB was dilated with a scale factor of 0.15 using point Q as the center of dilation, resulting in a smaller line segment A'B'. We were given the length of QA' as 2.4 units and asked to find the length of A'A, denoted as x. To solve this, we first established the relationship between the distances QA' and QA based on the scale factor, which is QA' = 0.15 * QA. By substituting the given value of QA' and solving for QA, we found that the length of QA is 16 units. Then, we used the fact that QA is the sum of QA' and A'A (i.e., QA = QA' + A'A) to set up an equation to solve for x. Substituting the known values, we had 16 = 2.4 + x. Solving this equation, we found that x = 13.6 units. This result provides a clear understanding of how dilation affects the lengths and positions of line segments. The dilated point A' is closer to the center of dilation Q than the original point A, and the distance between A' and A is 13.6 units. This problem highlights the practical application of dilation concepts in geometry and demonstrates how scale factors influence the size and position of geometric figures after transformation. Understanding these principles is essential for various applications, including geometric proofs, coordinate geometry, and real-world scaling problems in fields like architecture and design. The systematic approach used to solve this problem can be applied to a wide range of dilation-related problems, reinforcing the importance of mastering these fundamental concepts in geometry. This thorough analysis not only answers the specific question but also provides a comprehensive understanding of the underlying geometric principles and their applications.