Solving Coordinate Geometry Problem Finding Y-intercept And Line Equation

Jul 9, 2025 by Admin 74 views

Understanding Parallel Lines AB and CD Coordinate Geometry Problem

In coordinate geometry, the concepts of parallel lines, slopes, y-intercepts, and equations of lines are fundamental. This article delves into a problem involving parallel lines AB and CD, exploring how to find the y-intercept of line AB and the equation of line CD given specific coordinate points. We will break down the problem step-by-step, providing a comprehensive understanding of the underlying principles and calculations.

Problem Statement: Decoding the Geometry

The problem presents us with two parallel lines, AB and CD. We are given the coordinates of points A (8,0) and B (3,7) on line AB, and the coordinates of point D (5,5) on line CD. Our objectives are twofold:

Determine the y-intercept of line AB.
Find the equation of line CD.

This problem integrates several key concepts in coordinate geometry, including calculating the slope of a line, using the slope-intercept form of a linear equation, and applying the property that parallel lines have equal slopes. Let’s embark on a detailed journey to solve this problem.

Finding the y-intercept of AB: A Step-by-Step Guide

The y-intercept of a line is the point where the line crosses the y-axis. To find this point for line AB, we first need to determine the equation of the line. The slope-intercept form of a linear equation is given by:

y = mx + b

where:

y is the y-coordinate
x is the x-coordinate
m is the slope of the line
b is the y-intercept

Step 1: Calculate the Slope of AB

The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:

m = (y2 - y1) / (x2 - x1)

For line AB, the coordinates are A (8,0) and B (3,7). Plugging these values into the slope formula, we get:

m = (7 - 0) / (3 - 8) = 7 / -5 = -7/5

Thus, the slope of line AB is -7/5. Understanding the concept of slope is very important here. The slope defines how steep the line is and its direction. A negative slope, as in this case, indicates that the line goes downward from left to right. The magnitude of the slope (7/5) tells us that for every 5 units we move to the right on the x-axis, the line goes down by 7 units on the y-axis. This understanding helps in visualizing the line and predicting its behavior on the coordinate plane.

Step 2: Use the Point-Slope Form

Now that we have the slope, we can use the point-slope form of a line to find the equation. The point-slope form is given by:

y - y1 = m(x - x1)

where (x1, y1) is a point on the line. We can use either point A (8,0) or point B (3,7). Let’s use point A (8,0):

y - 0 = (-7/5)(x - 8)

Simplifying this, we get:

y = (-7/5)x + (7/5)*8 y = (-7/5)x + 56/5

The point-slope form is an invaluable tool in coordinate geometry because it allows us to write the equation of a line if we know the slope and one point on the line. This form is particularly useful when we don't immediately know the y-intercept, but we do have other information that can help us determine the line's equation. The transformation from the point-slope form to the slope-intercept form is a standard algebraic manipulation that further simplifies the equation and allows us to easily identify the y-intercept.

Step 3: Identify the y-intercept

Comparing the equation y = (-7/5)x + 56/5 with the slope-intercept form y = mx + b, we can see that the y-intercept b is 56/5.

Therefore, the y-intercept of line AB is 56/5, or 11.2 in decimal form. This result tells us exactly where the line AB intersects the y-axis. At the point (0, 11.2), the line AB crosses the vertical axis on the coordinate plane. This is a crucial piece of information as it helps us to fully define the line and visualize its position in the coordinate system. The y-intercept, along with the slope, provides a complete picture of the line’s characteristics and its relation to the coordinate axes.

Determining the Equation of Line CD: Leveraging Parallelism

To find the equation of line CD, we need to use the given information that AB and CD are parallel. Parallel lines have the same slope. Since we already found the slope of AB to be -7/5, the slope of CD will also be -7/5.

Step 1: Use the Slope and Point D

We know the slope of CD is -7/5, and we are given the coordinates of point D (5,5). We can use the point-slope form again:

y - y1 = m(x - x1)

Plugging in the values for point D (5,5) and the slope -7/5, we get:

y - 5 = (-7/5)(x - 5)

The concept of parallelism is a cornerstone in geometry, and its application in coordinate geometry problems is quite common. The fact that parallel lines share the same slope simplifies the process of finding the equation of a line significantly. Knowing the slope and a single point on the line is sufficient to define the line completely, and this is where the point-slope form proves to be incredibly useful. By using the slope derived from the parallel line AB and the coordinates of point D, we are able to set up the equation for line CD.

Step 2: Convert to Slope-Intercept Form

Now, let’s simplify the equation to the slope-intercept form y = mx + b:

y - 5 = (-7/5)x + (7/5)*5 y - 5 = (-7/5)x + 7 y = (-7/5)x + 7 + 5 y = (-7/5)x + 12

Converting the equation to slope-intercept form not only simplifies the equation but also provides immediate insights into the line’s properties. The slope-intercept form, y = mx + b, clearly displays the slope m and the y-intercept b, making it easier to visualize and analyze the line's behavior. In our case, after simplifying the equation, we arrive at the slope-intercept form for line CD, which gives us both the slope (which we already knew from the parallel line AB) and the y-intercept, thereby fully defining the line.

Step 3: Final Equation of Line CD

The equation of line CD in slope-intercept form is:

y = (-7/5)x + 12

This equation tells us that line CD has a slope of -7/5 and a y-intercept of 12. This means the line intersects the y-axis at the point (0, 12). The final equation succinctly captures the essence of line CD, describing its orientation and position in the coordinate plane. The slope of -7/5 indicates the line’s steepness and direction, while the y-intercept of 12 pinpoints exactly where the line crosses the y-axis. With this equation, we have a complete and clear mathematical representation of line CD.

Conclusion: Mastering Coordinate Geometry

In this article, we successfully determined the y-intercept of line AB (which is 56/5 or 11.2) and found the equation of line CD to be y = (-7/5)x + 12. This problem showcased the importance of understanding and applying fundamental concepts in coordinate geometry. The slope formula, point-slope form, slope-intercept form, and the properties of parallel lines are essential tools in solving such problems.

By breaking down the problem into manageable steps, we were able to systematically find the required solutions. This approach not only helps in solving specific problems but also builds a solid foundation for tackling more complex challenges in mathematics and related fields.

Mastering coordinate geometry involves understanding how algebraic equations translate into geometric shapes and positions on a plane. Problems like the one discussed here serve as excellent exercises to hone these skills. As we've demonstrated, a methodical approach combined with a clear understanding of fundamental concepts can lead to successful problem-solving in coordinate geometry.