A) Calculate The Distance Between Point D And Point E. B) Determine The Equation Of The Line DE. C) Find The Slope Of The Line EF. D) Determine The Coordinates Of The Midpoint Of The DF Segment.
In the realm of analytic geometry, triangles hold a fundamental position, serving as building blocks for more complex geometric figures and providing a rich context for exploring concepts such as distance, equations of lines, slopes, and midpoints. This article delves into the intricacies of triangle DEF, defined by the points D(-1,-3), E(-4,-2), and F(4,1) within the Cartesian coordinate system xOy. We will embark on a comprehensive exploration, systematically determining key properties of this triangle, including the distance between points, the equation of a line, the slope of a line, and the coordinates of the midpoint of a segment. Our journey will not only provide a step-by-step solution to each problem but also offer valuable insights into the underlying principles of analytic geometry, empowering you to tackle similar challenges with confidence.
Cartesian coordinates provide a powerful framework for representing and manipulating geometric objects algebraically. By associating points in the plane with ordered pairs of real numbers, we can leverage the tools of algebra to solve geometric problems and vice versa. This synergy between algebra and geometry is at the heart of analytic geometry, allowing us to express geometric concepts in a precise and quantitative manner. In the case of triangle DEF, the coordinates of its vertices, D(-1,-3), E(-4,-2), and F(4,1), serve as the foundation for our analysis. These coordinates encode the triangle's position and orientation within the Cartesian plane, enabling us to calculate distances, slopes, and other geometric properties.
Understanding the properties of triangle DEF not only provides a concrete example of geometric problem-solving but also reinforces fundamental concepts in analytic geometry. By calculating the distance between points D and E, we solidify our understanding of the distance formula, a cornerstone of analytic geometry. Determining the equation of line DE reinforces our knowledge of linear equations and their various forms, such as slope-intercept form and point-slope form. Calculating the slope of line EF allows us to explore the concept of inclination and the relationship between slope and the angle a line makes with the x-axis. Finally, finding the coordinates of the midpoint of a segment provides a practical application of the midpoint formula, a valuable tool for bisecting segments and finding their centers. Through this comprehensive analysis of triangle DEF, we aim to enhance your understanding of analytic geometry and equip you with the skills to tackle a wide range of geometric problems.
To determine the distance between two points in the Cartesian plane, we employ the distance formula, a direct application of the Pythagorean theorem. The distance formula states that the distance between two points (x1, y1) and (x2, y2) is given by:
distance = √((x2 - x1)² + (y2 - y1)²)
In our case, we want to find the distance between point D(-1,-3) and point E(-4,-2). Let's designate D as (x1, y1) and E as (x2, y2). Plugging the coordinates into the distance formula, we get:
distance(DE) = √((-4 - (-1))² + (-2 - (-3))²)
Simplifying the expression inside the square root, we have:
distance(DE) = √((-4 + 1)² + (-2 + 3)²)
distance(DE) = √((-3)² + (1)²)
distance(DE) = √(9 + 1)
distance(DE) = √10
Therefore, the distance from point D to point E is √10 units. This calculation demonstrates the power of the distance formula in quantifying the separation between two points in the Cartesian plane. The distance formula is a fundamental tool in analytic geometry, used extensively in various applications, including finding the lengths of line segments, determining the perimeter of polygons, and calculating the distance between a point and a line. Understanding and applying the distance formula is crucial for mastering analytic geometry and solving a wide range of geometric problems.
The result, √10, is an irrational number, meaning it cannot be expressed as a simple fraction. This highlights the importance of working with both rational and irrational numbers in analytic geometry. While we can approximate √10 as a decimal (approximately 3.16), the exact value is √10, which provides the most accurate representation of the distance between points D and E. This emphasizes the significance of preserving exact values whenever possible in mathematical calculations, especially when dealing with irrational numbers. The distance formula, in conjunction with the understanding of rational and irrational numbers, forms a cornerstone of analytic geometry, enabling us to precisely measure distances and explore geometric relationships within the Cartesian plane.
The application of the distance formula to find the distance between points D and E serves as a fundamental step in analyzing triangle DEF. This distance represents the length of one of the triangle's sides, a crucial parameter for further calculations such as finding the perimeter of the triangle or determining its area. The distance between two points is not merely a numerical value; it carries geometric significance, representing the physical separation between the points in the plane. By accurately calculating the distance between points D and E, we gain a deeper understanding of the triangle's dimensions and its position within the Cartesian coordinate system. This understanding lays the groundwork for exploring other properties of the triangle, such as the equations of its sides, the measures of its angles, and its overall shape and characteristics. The distance formula, therefore, is not just a mathematical tool but a gateway to unraveling the geometric properties of figures within the Cartesian plane.
To find the equation of the line passing through points D(-1,-3) and E(-4,-2), we can use the point-slope form of a linear equation. This form is particularly useful when we know a point on the line and its slope. The point-slope form is given by:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line and m is the slope of the line.
First, we need to calculate the slope (m) of line DE. The slope is defined as the change in y divided by the change in x:
m = (y2 - y1) / (x2 - x1)
Using the coordinates of points D and E, we have:
m = (-2 - (-3)) / (-4 - (-1))
m = (-2 + 3) / (-4 + 1)
m = 1 / -3
m = -1/3
Now that we have the slope, m = -1/3, we can use the point-slope form. We can choose either point D or point E as (x1, y1). Let's use point D(-1,-3):
y - (-3) = (-1/3)(x - (-1))
y + 3 = (-1/3)(x + 1)
To get the equation in slope-intercept form (y = mx + b), we can simplify and solve for y:
y + 3 = (-1/3)x - 1/3
y = (-1/3)x - 1/3 - 3
y = (-1/3)x - 1/3 - 9/3
y = (-1/3)x - 10/3
Alternatively, we can express the equation in the general form (Ax + By + C = 0) by multiplying both sides by 3 and rearranging:
3y = -x - 10
x + 3y + 10 = 0
Therefore, the equation of line DE can be expressed in slope-intercept form as y = (-1/3)x - 10/3 or in general form as x + 3y + 10 = 0. Determining the equation of a line is a fundamental skill in analytic geometry, allowing us to represent linear relationships algebraically and analyze their properties. The point-slope form provides a convenient method for finding the equation of a line when we know a point on the line and its slope, while the slope-intercept form and general form offer alternative representations that are useful in different contexts.
The ability to represent a line using an algebraic equation is a cornerstone of analytic geometry, bridging the gap between geometric concepts and algebraic expressions. The equation of a line encapsulates all the points that lie on the line, providing a concise and powerful way to describe its position and orientation within the Cartesian plane. By finding the equation of line DE, we have effectively captured its essence in a mathematical formula, allowing us to perform further analysis, such as finding its intersections with other lines or determining its distance from a point. The equation of a line is not just a formula; it is a tool for understanding and manipulating linear relationships in a geometric context.
The process of determining the equation of line DE highlights the interconnectedness of various concepts in analytic geometry. We first calculated the slope of the line, a measure of its steepness and direction. Then, we utilized the point-slope form, a versatile tool that allows us to construct the equation of a line given a point and its slope. Finally, we transformed the equation into slope-intercept form and general form, demonstrating the different ways in which a linear relationship can be represented algebraically. This journey through different forms of linear equations underscores the flexibility and power of analytic geometry in expressing and analyzing geometric relationships. The equation of line DE, therefore, is not just a final answer but a culmination of various geometric and algebraic concepts working in harmony.
To determine the slope of the line passing through points E(-4,-2) and F(4,1), we again use the formula for slope:
m = (y2 - y1) / (x2 - x1)
In this case, let E be (x1, y1) and F be (x2, y2). Plugging the coordinates into the formula, we get:
m = (1 - (-2)) / (4 - (-4))
m = (1 + 2) / (4 + 4)
m = 3 / 8
Therefore, the slope of line EF is 3/8. The slope is a crucial property of a line, indicating its direction and steepness. A positive slope, such as 3/8, signifies that the line is increasing as we move from left to right in the Cartesian plane. The magnitude of the slope indicates the rate of increase; a larger slope corresponds to a steeper line, while a smaller slope corresponds to a gentler incline. The slope of a line is a fundamental concept in analytic geometry, used extensively in various applications, including determining the angle a line makes with the x-axis, finding the equations of parallel and perpendicular lines, and analyzing the behavior of linear functions.
The value of the slope, 3/8, provides valuable insights into the orientation of line EF within the Cartesian plane. The fact that the slope is positive indicates that the line rises as we move from left to right. For every 8 units we move horizontally, the line rises 3 units vertically. This visual representation of the slope helps us to understand the line's inclination and its relationship to the coordinate axes. A line with a slope of 3/8 is less steep than a line with a slope of, say, 1, but steeper than a line with a slope of 1/4. Understanding the magnitude and sign of the slope is essential for interpreting the geometric properties of a line and its behavior in the Cartesian plane.
The calculation of the slope of line EF contributes to our overall understanding of triangle DEF. The slopes of the triangle's sides provide information about its angles and shape. For instance, if two lines are perpendicular, their slopes are negative reciprocals of each other. By comparing the slopes of lines DE and EF, we can gain insights into the angles formed within the triangle. The slope is not just a numerical value; it is a geometric descriptor that helps us to visualize and analyze the relationships between lines and shapes within the Cartesian plane. The slope of line EF, therefore, is a key piece of information that helps us to characterize the properties of triangle DEF and its position within the coordinate system.
The midpoint of a line segment is the point that divides the segment into two equal parts. To find the coordinates of the midpoint of segment DF, we use the midpoint formula, which states that the coordinates of the midpoint (xm, ym) of a segment with endpoints (x1, y1) and (x2, y2) are given by:
xm = (x1 + x2) / 2
ym = (y1 + y2) / 2
In our case, we want to find the midpoint of segment DF, where D(-1,-3) and F(4,1). Plugging the coordinates into the midpoint formula, we get:
xm = (-1 + 4) / 2
xm = 3 / 2
ym = (-3 + 1) / 2
ym = -2 / 2
ym = -1
Therefore, the coordinates of the midpoint of segment DF are (3/2, -1). The midpoint is a significant point on a line segment, representing its center and dividing it into two equal halves. The midpoint formula provides a straightforward method for calculating the coordinates of this point, given the coordinates of the endpoints of the segment. Finding the midpoint has various applications in geometry, including bisecting segments, finding the center of a circle, and determining the medians of a triangle. The midpoint formula is a valuable tool in analytic geometry, enabling us to locate and analyze the central points of line segments and geometric figures.
The coordinates of the midpoint of segment DF, (3/2, -1), provide a precise location for the center of the segment within the Cartesian plane. The x-coordinate, 3/2, indicates that the midpoint lies 1.5 units to the right of the y-axis, while the y-coordinate, -1, indicates that it lies 1 unit below the x-axis. This precise location allows us to visualize the midpoint's position relative to the endpoints of the segment and to the coordinate axes. Understanding the coordinates of the midpoint is crucial for accurately representing and manipulating geometric figures in the Cartesian plane. The midpoint is not just a point; it is a reference point that helps us to understand the properties and relationships of line segments and geometric shapes.
The determination of the midpoint of segment DF contributes to our overall understanding of triangle DEF. The midpoint of a side is an important point in the context of triangle geometry, particularly when considering concepts such as medians and centroids. A median of a triangle is a line segment that connects a vertex to the midpoint of the opposite side. The centroid of a triangle, the point where the three medians intersect, is the triangle's center of gravity. By finding the midpoint of segment DF, we have taken a step towards exploring these concepts and further characterizing the properties of triangle DEF. The midpoint, therefore, is not just a point on a line segment; it is a gateway to exploring more advanced geometric concepts and relationships within triangles and other geometric figures.
In this comprehensive exploration of triangle DEF, we have successfully determined several key properties, including the distance between points D and E, the equation of line DE, the slope of line EF, and the coordinates of the midpoint of segment DF. These calculations demonstrate the power of analytic geometry in solving geometric problems using algebraic techniques. By applying the distance formula, the point-slope form of a linear equation, the slope formula, and the midpoint formula, we have gained a deeper understanding of triangle DEF and its characteristics within the Cartesian coordinate system. This analysis not only provides specific answers to the given questions but also reinforces fundamental concepts in analytic geometry, equipping you with the skills to tackle similar problems with confidence.
The process of analyzing triangle DEF has highlighted the interconnectedness of various concepts in analytic geometry. We have seen how the distance formula, the slope formula, and the midpoint formula work together to provide a comprehensive understanding of geometric figures. The equation of a line, derived using the point-slope form, encapsulates the relationship between the x and y coordinates of all points on the line. The slope, a fundamental property of a line, indicates its direction and steepness. The midpoint, a central point on a line segment, divides the segment into two equal halves. By mastering these concepts and their interrelationships, you can effectively analyze and manipulate geometric figures within the Cartesian plane.
This exploration of triangle DEF serves as a foundation for further studies in analytic geometry and related fields. The concepts and techniques we have discussed are applicable to a wide range of geometric problems, including finding the areas of polygons, determining the equations of circles and other conic sections, and analyzing the transformations of geometric figures. By building a strong foundation in analytic geometry, you will be well-prepared to tackle more advanced topics in mathematics and to apply these concepts to real-world problems in fields such as engineering, physics, and computer graphics. The journey through triangle DEF is not just an exercise in problem-solving; it is a stepping stone towards a deeper understanding of mathematics and its applications.