Finding The Equation Of Line BC A Geometry Problem Solution

JU07/11/2025 08, 2025 by THE IDEN 60 views

Unlocking the Equation of 𝐁𝐂: A Comprehensive Guide to Solving the Geometry Problem

In the realm of geometry, pinpointing the equation of a line given specific conditions is a fundamental skill. This article serves as an exhaustive guide, meticulously dissecting a geometry problem to elucidate the process of determining the equation of a line. We will delve into the problem, focusing on the interplay between perpendicular lines and their slopes. By unraveling the solution step-by-step, this guide aims to enhance your understanding and proficiency in tackling similar geometric challenges. So, let's embark on this mathematical journey and unlock the equation of 𝐁𝐂!

Problem Statement: Decoding the Geometric Puzzle

Let's begin by stating the problem clearly:

Problem: Lines $\overleftrightarrow{A B}$ and $\overleftrightarrow{B C}$ intersect to form a right angle at point $B$ . Given the coordinates $A=(-3,-1)$ and $B=(4,4)$ , what is the equation of line $\overrightarrow{B C}$ ? The options are:

A. $x+3 y=16$ B. $2 x+y=12$ C. $-7 x-5 y=-48$ D. $7x-5y=8$

This problem presents us with a classic geometric scenario involving perpendicular lines. Our mission is to decipher the relationship between the given points and the right angle to deduce the equation of the line $\overrightarrow{B C}$ . To do so effectively, we'll dissect the information provided and systematically employ geometric principles to arrive at the correct answer.

Solution: A Step-by-Step Geometric Expedition

To solve this problem, we'll embark on a step-by-step expedition, employing the principles of coordinate geometry and the properties of perpendicular lines. Our journey will involve calculating slopes, applying the condition for perpendicularity, and finally, constructing the equation of the line $\overrightarrow{B C}$ . Let's begin this geometric adventure!

1. Calculating the Slope of 𝐀𝐁: The Foundation of Our Solution

In the initial phase of our solution, the crucial step involves determining the slope of line $\overleftrightarrow{A B}$ . Given the coordinates of points $A=(-3,-1)$ and $B=(4,4)$ , we can calculate the slope ( $m_{AB}$ ) using the slope formula:

$m_{AB} = \frac{y_2 - y_1}{x_2 - x_1}$

Substituting the coordinates of points A and B into the formula, we get:

$m_{AB} = \frac{4 - (-1)}{4 - (-3)} = \frac{5}{7}$

Thus, the slope of line $\overleftrightarrow{A B}$ is $\frac{5}{7}$ . This value lays the foundation for our subsequent steps, enabling us to utilize the properties of perpendicular lines to unveil the slope of line $\overrightarrow{B C}$ .

2. Utilizing Perpendicularity: Unveiling the Slope of 𝐁𝐂

At the core of this problem lies the concept of perpendicular lines. Since $\overleftrightarrow{A B}$ and $\overleftrightarrow{B C}$ form a right angle at point B, they are perpendicular to each other. A fundamental property of perpendicular lines states that the product of their slopes is -1. In mathematical terms:

$m_{AB} \cdot m_{BC} = -1$

We have already determined that the slope of $\overleftrightarrow{A B}$ , $m_{AB}$ , is $\frac{5}{7}$ . Now, we can use this information to find the slope of $\overrightarrow{B C}$ , $m_{BC}$ . Substituting the value of $m_{AB}$ into the equation, we get:

$\frac{5}{7} \cdot m_{BC} = -1$

To solve for $m_{BC}$ , we multiply both sides of the equation by the reciprocal of $\frac{5}{7}$ , which is $\frac{7}{5}$ :

$m_{BC} = -1 \cdot \frac{7}{5} = -\frac{7}{5}$

Therefore, the slope of line $\overrightarrow{B C}$ is $-\frac{7}{5}$ . This crucial piece of information brings us closer to defining the equation of the line.

3. Constructing the Equation of 𝐁𝐂: The Final Revelation

Having determined the slope of line $\overrightarrow{B C}$ as $-\frac{7}{5}$ , we now possess the necessary ingredients to construct its equation. We know that line $\overrightarrow{B C}$ passes through point $B=(4,4)$ . To formulate the equation, we will employ the point-slope form, a powerful tool in coordinate geometry. The point-slope form is defined as:

$y - y_1 = m(x - x_1)$

where $(x_1, y_1)$ represents a point on the line, and $m$ denotes the slope of the line.

In our case, we have the point $B=(4,4)$ and the slope $m_{BC} = -\frac{7}{5}$ . Substituting these values into the point-slope form, we get:

$y - 4 = -\frac{7}{5}(x - 4)$

To eliminate the fraction and express the equation in a more standard form, we multiply both sides of the equation by 5:

$5(y - 4) = -7(x - 4)$

Expanding both sides, we have:

$5y - 20 = -7x + 28$

Now, let's rearrange the terms to obtain the equation in the general form, which is $Ax + By = C$ :

$7x + 5y = 48$

Multiplying both sides by -1, we get:

$-7x - 5y = -48$

Comparing this equation with the given options, we find that it matches option C.

4. Conclusion: Validating Our Geometric Solution

Through a meticulous step-by-step process, we have successfully navigated the geometric puzzle and arrived at the equation of line $\overrightarrow{B C}$ . Our journey involved calculating slopes, applying the principle of perpendicularity, and employing the point-slope form to construct the equation. We found that the equation of line $\overrightarrow{B C}$ is:

$-7x - 5y = -48$

This result aligns with option C, thus confirming our solution. By systematically dissecting the problem and applying the fundamental principles of coordinate geometry, we have not only solved the problem but also reinforced our understanding of geometric concepts. This experience serves as a testament to the power of analytical problem-solving in the realm of mathematics.

Final Answer: The Geometric Revelation

Therefore, the correct answer is:

C. $-7 x-5 y=-48$

This comprehensive solution not only provides the answer but also illuminates the underlying geometric principles and problem-solving techniques. By grasping these concepts, you'll be well-equipped to tackle similar challenges in the fascinating world of geometry.

Keywords and Concepts: Building Blocks of Geometric Understanding

Before we conclude, let's reinforce our understanding by summarizing the key keywords and concepts that played pivotal roles in solving this problem. These building blocks of geometric understanding will serve as valuable tools in your future mathematical endeavors.

Slope of a line: The slope measures the steepness and direction of a line. It is calculated as the change in y divided by the change in x.
Perpendicular lines: Perpendicular lines intersect at a right angle (90 degrees). The product of their slopes is -1.
Point-slope form: The point-slope form is a convenient way to express the equation of a line when you know a point on the line and its slope.
Equation of a line: The equation of a line is a mathematical expression that describes the relationship between the x and y coordinates of points on the line.
Coordinate geometry: Coordinate geometry is a branch of mathematics that deals with the study of geometric figures using a coordinate system.

By mastering these keywords and concepts, you'll be well-prepared to tackle a wide array of geometric problems. Remember, practice is key to solidifying your understanding and enhancing your problem-solving skills.

FAQs: Addressing Common Queries in Geometry

To further solidify your understanding, let's address some frequently asked questions related to this problem and the concepts involved. These FAQs will provide additional insights and clarify any lingering queries you may have.

Q1: How do you find the slope of a line given two points?

To find the slope of a line given two points, you can use the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ , where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two points.

Q2: What is the relationship between the slopes of perpendicular lines?

The slopes of perpendicular lines are negative reciprocals of each other. This means that if one line has a slope of $m$ , the slope of a line perpendicular to it will be $-\frac{1}{m}$ .

Q3: Can you explain the point-slope form of a line?

The point-slope form of a line is given by $y - y_1 = m(x - x_1)$ , where $(x_1, y_1)$ is a point on the line and $m$ is the slope of the line. This form is particularly useful when you know a point on the line and its slope.

Q4: How do you convert an equation from point-slope form to general form?

To convert an equation from point-slope form to general form ( $Ax + By = C$ ), you need to expand the equation, eliminate fractions (if any), and rearrange the terms so that the equation is in the desired form.

Q5: Why is it important to understand the concept of slope in geometry?

The concept of slope is fundamental in geometry because it provides information about the steepness and direction of a line. It is crucial for determining the relationship between lines (parallel, perpendicular, etc.) and for writing the equations of lines.

By addressing these frequently asked questions, we aim to provide a comprehensive understanding of the concepts involved and equip you with the knowledge to confidently tackle similar problems in the future.

Practice Problems: Sharpening Your Geometric Acumen

To truly master the concepts and techniques discussed in this guide, practice is essential. Here are a few practice problems that will challenge your understanding and sharpen your geometric acumen. Work through these problems, applying the principles we've covered, and watch your problem-solving skills soar!

Line $\overleftrightarrow{P Q}$ has points $P(2, 3)$ and $Q(5, 7)$ . Line $\overleftrightarrow{R S}$ is perpendicular to $\overleftrightarrow{P Q}$ and passes through point $R(1, 4)$ . Find the equation of line $\overleftrightarrow{R S}$ .
A line has a slope of -2 and passes through the point (3, -1). Write the equation of the line in slope-intercept form ( $y = mx + b$ ).
Determine whether the lines $2x + 3y = 6$ and $3x - 2y = 4$ are perpendicular.

By tackling these practice problems, you'll solidify your understanding of the concepts and techniques discussed in this guide. Remember, consistent practice is the key to mastering any mathematical skill. So, dive in, challenge yourself, and watch your geometric prowess flourish!

Conclusion: Empowering Your Geometric Journey

In conclusion, this comprehensive guide has meticulously dissected the problem of finding the equation of a line, focusing on the interplay between perpendicular lines and their slopes. We embarked on a step-by-step journey, starting with calculating slopes, applying the condition for perpendicularity, and culminating in the construction of the equation of line $\overrightarrow{B C}$ .

By understanding the underlying concepts, mastering the problem-solving techniques, and consistently practicing, you can confidently navigate the world of geometry. We encourage you to continue exploring, challenging yourself, and pushing the boundaries of your mathematical knowledge. The journey of learning is an ongoing adventure, and we hope this guide has empowered you to embrace it with enthusiasm and confidence.