function calculateSlope(x1, y1, x2, y2) {
var dx = x2 – x1;
var dy = y2 – y1;
if (dx === 0) {
return Infinity; // Represents an undefined slope (vertical line)
}
return dy / dx;
}
function calculateLineRelationship() {
// Get values for Line 1
var x1_line1 = parseFloat(document.getElementById("x1_line1").value);
var y1_line1 = parseFloat(document.getElementById("y1_line1").value);
var x2_line1 = parseFloat(document.getElementById("x2_line1").value);
var y2_line1 = parseFloat(document.getElementById("y2_line1").value);
// Get values for Line 2
var x1_line2 = parseFloat(document.getElementById("x1_line2").value);
var y1_line2 = parseFloat(document.getElementById("y1_line2").value);
var x2_line2 = parseFloat(document.getElementById("x2_line2").value);
var y2_line2 = parseFloat(document.getElementById("y2_line2").value);
// Validate inputs
if (isNaN(x1_line1) || isNaN(y1_line1) || isNaN(x2_line1) || isNaN(y2_line1) ||
isNaN(x1_line2) || isNaN(y1_line2) || isNaN(x2_line2) || isNaN(y2_line2)) {
document.getElementById("slope1Result").textContent = "Invalid input";
document.getElementById("slope2Result").textContent = "Invalid input";
document.getElementById("relationshipResult").textContent = "Please enter valid numbers for all coordinates.";
return;
}
// Calculate slopes
var m1 = calculateSlope(x1_line1, y1_line1, x2_line1, y2_line1);
var m2 = calculateSlope(x1_line2, y1_line2, x2_line2, y2_line2);
var relationship = "";
// Display slopes
document.getElementById("slope1Result").textContent = (m1 === Infinity) ? "Undefined (Vertical Line)" : m1.toFixed(4);
document.getElementById("slope2Result").textContent = (m2 === Infinity) ? "Undefined (Vertical Line)" : m2.toFixed(4);
// Determine relationship
if (m1 === Infinity && m2 === Infinity) {
relationship = "Parallel (Both are vertical lines)";
} else if (m1 === 0 && m2 === 0) {
relationship = "Parallel (Both are horizontal lines)";
} else if (m1 === m2) {
relationship = "Parallel";
} else if ((m1 === Infinity && m2 === 0) || (m1 === 0 && m2 === Infinity)) {
relationship = "Perpendicular";
} else if (m1 !== Infinity && m2 !== Infinity && Math.abs(m1 * m2 + 1) < 1e-9) { // Using a small epsilon for floating point comparison
relationship = "Perpendicular";
} else {
relationship = "Neither Parallel nor Perpendicular";
}
document.getElementById("relationshipResult").textContent = relationship;
}
.calculator-container {
background-color: #f9f9f9;
border: 1px solid #ddd;
padding: 20px;
border-radius: 8px;
max-width: 600px;
margin: 20px auto;
font-family: Arial, sans-serif;
}
.calculator-container h2, .calculator-container h3 {
color: #333;
text-align: center;
margin-bottom: 15px;
}
.calc-input-group {
margin-bottom: 15px;
padding: 10px;
border: 1px solid #eee;
border-radius: 5px;
background-color: #fff;
}
.calc-input-group label {
display: block;
margin-bottom: 5px;
color: #555;
font-weight: bold;
}
.calc-input-group input[type="number"] {
width: calc(100% – 12px);
padding: 8px;
margin-bottom: 10px;
border: 1px solid #ccc;
border-radius: 4px;
box-sizing: border-box;
}
.calc-button {
display: block;
width: 100%;
padding: 10px 15px;
background-color: #007bff;
color: white;
border: none;
border-radius: 4px;
font-size: 16px;
cursor: pointer;
transition: background-color 0.3s ease;
margin-top: 20px;
}
.calc-button:hover {
background-color: #0056b3;
}
.calc-results {
margin-top: 20px;
padding: 15px;
border: 1px solid #e0e0e0;
border-radius: 5px;
background-color: #e9ecef;
}
.calc-results p {
margin-bottom: 8px;
color: #333;
font-size: 1.1em;
}
.calc-results span {
font-weight: bold;
color: #007bff;
}
Understanding Parallel, Perpendicular, and Neither Lines
In geometry, the relationship between two lines in a plane can be categorized into three main types: parallel, perpendicular, or neither. This classification is primarily determined by their slopes.
What is Slope?
The slope of a line is a measure of its steepness. It describes how much the line rises or falls for a given horizontal distance. Mathematically, the slope (often denoted by 'm') of a line passing through two points (x1, y1) and (x2, y2) is calculated using the formula:
m = (y2 - y1) / (x2 - x1)
A positive slope indicates an upward trend, a negative slope indicates a downward trend, a slope of zero indicates a horizontal line, and an undefined slope (when x2 – x1 = 0) indicates a vertical line.
Parallel Lines
Two distinct lines are considered parallel if they lie in the same plane and never intersect. This occurs when they have the exact same slope. If line 1 has slope m1 and line 2 has slope m2, then:
m1 = m2
Special Case: Two vertical lines are always parallel to each other, as both have undefined slopes.
Example: A line passing through (1, 2) and (3, 4) has a slope of (4-2)/(3-1) = 2/2 = 1. Another line passing through (0, 5) and (2, 7) has a slope of (7-5)/(2-0) = 2/2 = 1. Since their slopes are equal, these lines are parallel.
Perpendicular Lines
Two lines are perpendicular if they intersect at a right angle (90 degrees). This relationship is defined by their slopes having a negative reciprocal relationship. If line 1 has slope m1 and line 2 has slope m2, then:
m1 * m2 = -1
This also means that m2 = -1 / m1 (provided m1 is not zero).
Special Case: A horizontal line (slope = 0) is perpendicular to a vertical line (undefined slope). Their product cannot be -1, but this is a fundamental perpendicular relationship.
Example: A line passing through (1, 2) and (3, 3) has a slope of (3-2)/(3-1) = 1/2. Another line passing through (0, 5) and (2, 1) has a slope of (1-5)/(2-0) = -4/2 = -2. Since (1/2) * (-2) = -1, these lines are perpendicular.
Neither Parallel nor Perpendicular
If two lines are not parallel (their slopes are not equal) and not perpendicular (the product of their slopes is not -1, and they are not a horizontal/vertical pair), then they are classified as neither parallel nor perpendicular. These lines will intersect at some point, but not at a 90-degree angle.
Example: A line passing through (1, 2) and (3, 5) has a slope of (5-2)/(3-1) = 3/2. Another line passing through (0, 0) and (2, 3) has a slope of (3-0)/(2-0) = 3/2. These lines are parallel. If the second line passed through (0,0) and (2,1), its slope would be 1/2. Since 3/2 is not equal to 1/2, and (3/2)*(1/2) = 3/4 which is not -1, these lines would be neither parallel nor perpendicular.
How to Use the Calculator
To use the calculator, simply input the X and Y coordinates for two distinct points on Line 1, and then two distinct points for Line 2. The calculator will automatically compute the slopes of both lines and determine whether they are parallel, perpendicular, or neither, based on the rules described above.
Remember that for vertical lines, the slope is undefined. The calculator handles this special case to correctly identify parallel vertical lines or perpendicular horizontal/vertical line pairs.