Find Rectangle Vertices: Dimensions 1.2, 7.5, Vertex (7,2)

Hey everyone! Let's dive into a fun geometry problem today. We're going to figure out how to find the other vertices of a rectangle when we know one vertex and the dimensions of the rectangle. It might sound tricky, but trust me, we'll break it down into easy-to-follow steps. So, grab your thinking caps, and let's get started!

Understanding the Problem: Rectangles and Their Properties

Before we jump into the solution, let's quickly recap what makes a rectangle a rectangle. The key properties of a rectangle are that it's a four-sided shape (a quadrilateral) with four right angles (90-degree angles). This means opposite sides are parallel and equal in length. Understanding these basics is crucial because they'll guide us as we find the missing vertices. When approaching these geometric problems, it's always good to visualize. Think of rectangles you see every day – books, screens, doors – and remember their corners are perfectly square. We'll use these fundamental properties to solve our puzzle.

When we talk about the vertices of a rectangle, we're referring to the corner points where the sides meet. Each vertex is a specific location in a two-dimensional plane, defined by its x and y coordinates. For instance, the given vertex (7, 2) tells us one corner of our rectangle is located 7 units along the x-axis and 2 units along the y-axis. Our mission is to pinpoint the coordinates of the remaining three corners, using the dimensions and the properties of the rectangle. To start, let's remember that a rectangle has two dimensions: its length and its width. These dimensions dictate the distance between the sides and will be instrumental in locating the other vertices. Remember, rectangles aren't just random quadrilaterals; they have a structured elegance. Their right angles and parallel sides create a predictable framework that we can leverage to our advantage.

Setting Up the Scenario: Dimensions and the Known Vertex

In our specific scenario, we're dealing with a rectangle that has dimensions of 1.2 and 7.5 units. This means one side of the rectangle is 1.2 units long, and the other side is 7.5 units long. Now, it's important to remember that these dimensions can represent either the length or the width of the rectangle, depending on how it's oriented in the coordinate plane. We also know that one of the vertices of this rectangle is located at the point (7, 2). This is our anchor point, our starting point from which we'll determine the locations of the other three vertices. Think of it like having one corner of a puzzle already in place – we need to find the other three pieces that fit perfectly to complete the rectangle. To make this even clearer, it often helps to sketch a quick diagram. While it doesn't need to be perfectly to scale, a visual representation can give you a better sense of the rectangle's orientation and the relative positions of the vertices. This can prevent confusion and help you avoid making simple mistakes. So, let's imagine our rectangle with one corner at (7, 2), and sides of 1.2 and 7.5 units – what could the other corners be?

Finding the Adjacent Vertices: Using the Dimensions

Okay, guys, now comes the fun part – finding the other vertices! Since we know one vertex is at (7, 2) and we have the dimensions of the rectangle (1.2 and 7.5), we can start by finding the vertices that are directly adjacent to (7, 2). These are the vertices that share a side with our known vertex. Remember, the sides of a rectangle are perpendicular, meaning they form right angles. This is key because it tells us we can move either horizontally (changing the x-coordinate) or vertically (changing the y-coordinate) to find these adjacent vertices. Let's start by considering the side with a length of 1.2 units. We can move 1.2 units to the right or left of our known vertex (7, 2). Moving to the right would mean adding 1.2 to the x-coordinate, giving us a new vertex at (7 + 1.2, 2), which is (8.2, 2). Moving to the left would mean subtracting 1.2 from the x-coordinate, giving us a vertex at (7 - 1.2, 2), which is (5.8, 2).

Now, let's consider the other side with a length of 7.5 units. This time, we'll move vertically from our known vertex. Moving upwards means adding 7.5 to the y-coordinate, resulting in a vertex at (7, 2 + 7.5), which is (7, 9.5). Moving downwards means subtracting 7.5 from the y-coordinate, giving us a vertex at (7, 2 - 7.5), which is (7, -5.5). So far, we've found four potential adjacent vertices: (8.2, 2), (5.8, 2), (7, 9.5), and (7, -5.5). But remember, a rectangle only has three other vertices, not four. This means that only two of these points can be adjacent to (7, 2). This is where we need to think about which pair of vertices will form sides that are perpendicular to each other. Let’s examine these points more closely to determine the correct adjacent vertices.

Identifying the Correct Vertices: Perpendicularity is Key

We've got a list of potential vertices, but how do we know which ones are the real deal? This is where the perpendicularity of the sides of a rectangle comes into play. Remember, adjacent sides of a rectangle form right angles, meaning they are perpendicular. In terms of coordinates, this means that if we move horizontally to find one adjacent vertex, we must move vertically to find the other. We can't have two vertices that are both horizontally or both vertically aligned with our starting vertex. Let’s take a closer look at our potential vertices: (8.2, 2), (5.8, 2), (7, 9.5), and (7, -5.5). Notice that the vertices (8.2, 2) and (5.8, 2) both have the same y-coordinate as our starting vertex (7, 2). This means they are horizontally aligned with (7, 2). Similarly, the vertices (7, 9.5) and (7, -5.5) have the same x-coordinate as (7, 2), indicating they are vertically aligned. This gives us a crucial clue: one of our adjacent vertices must be horizontally aligned, and the other must be vertically aligned. So, we need to pick one vertex from the pair (8.2, 2) and (5.8, 2), and another from the pair (7, 9.5) and (7, -5.5). Let's say we choose (8.2, 2) as one adjacent vertex (moving 1.2 units to the right). Then, the other adjacent vertex must be one of the vertically aligned points, either (7, 9.5) or (7, -5.5). This is where we start to see how the geometry of the rectangle dictates the positions of its vertices. The perpendicularity rule acts like a filter, helping us narrow down the possibilities and pinpoint the correct locations.

Finding the Final Vertex: Completing the Rectangle

Alright, we've nailed down two vertices! Let’s say, for example, we've determined that the vertices adjacent to (7, 2) are (8.2, 2) and (7, 9.5). Now, to find the fourth and final vertex, we need to complete the rectangle. Think of it like connecting the dots – we have three corners, and we just need to find the last one that makes everything fit perfectly. To do this, we can use the fact that opposite sides of a rectangle are parallel and equal in length. This means that the side connecting our final vertex to (8.2, 2) must be parallel and equal in length to the side connecting (7, 2) and (7, 9.5). Similarly, the side connecting our final vertex to (7, 9.5) must be parallel and equal in length to the side connecting (7, 2) and (8.2, 2). Let's break this down into steps. To get from (7, 2) to (7, 9.5), we moved 7.5 units upwards (the y-coordinate increased by 7.5). So, to find our final vertex, we need to move 7.5 units upwards from (8.2, 2). This gives us a new y-coordinate of 2 + 7.5 = 9.5. Therefore, the final vertex will have a y-coordinate of 9.5. Next, to get from (7, 2) to (8.2, 2), we moved 1.2 units to the right (the x-coordinate increased by 1.2). So, we need to move 1.2 units to the right from (7, 9.5). This gives us a new x-coordinate of 7 + 1.2 = 8.2. Putting it all together, our final vertex is located at (8.2, 9.5). We've now successfully found all four vertices of the rectangle!

Summarizing the Solution and Key Takeaways

So, to recap, guys, we started with one vertex (7, 2) and the dimensions of the rectangle (1.2 and 7.5). By using the properties of rectangles – right angles, parallel sides, and equal lengths – we were able to systematically find the other three vertices. We first identified the adjacent vertices by moving horizontally and vertically from our known vertex, considering the dimensions of the rectangle. Then, we used the concept of perpendicularity to select the correct adjacent vertices. Finally, we completed the rectangle by using the properties of parallel and equal sides to find the location of the final vertex. The key takeaway here is that geometry problems often require us to combine different concepts and properties to arrive at the solution. Understanding the fundamental characteristics of shapes, like rectangles, is crucial. In this case, knowing that rectangles have right angles and parallel sides was the cornerstone of our approach. Another important skill is visualizing the problem. Sketching a quick diagram can often make the relationships between points and lines much clearer. Don't be afraid to draw things out! Finally, remember that there might be multiple approaches to solving a geometry problem. We chose one path in this example, but there could be other valid ways to find the vertices. The important thing is to understand the underlying principles and apply them logically. Keep practicing, and you'll become a rectangle-solving pro in no time!