To find an equation perpendicular to a line, determine its slope by calculating the negative reciprocal of the original line’s slope. Using a point on the perpendicular line, apply the point-slope form to derive its equation. Finally, simplify it into the slope-intercept form to obtain the equation perpendicular to the given line.

## Understanding Perpendicular Lines: A Foundation for Geometry

In the vast realm of geometry, understanding perpendicular lines holds immense significance. **Perpendicular lines** are two lines that intersect at right angles, forming a 90-degree angle at their point of intersection. This unique characteristic makes them essential for a wide range of applications, from architecture to engineering.

Lines, planes, and segments can all exhibit perpendicularity. When two **lines** intersect at right angles, they are considered perpendicular. Similarly, when two **planes** intersect to form a right angle, they are also perpendicular. Lastly, two **segments** that meet to form a right angle are considered perpendicular.

Understanding perpendicular lines is a crucial step in the journey of geometry. It provides a foundation for exploring concepts like parallel lines, slopes, and equations, which are vital for solving problems and understanding the intricate relationships between geometric figures.

**Slope and Intercept of a Line**

- What is slope and how to calculate it
- Y-intercept and x-intercept

**Slope and Intercept of a Line: Understanding the Building Blocks of Equations**

In the realm of geometry, lines play a pivotal role in shaping our understanding of space and relationships. Describing the direction and position of a line is essential for problem-solving and analysis. Two key concepts that unravel the mysteries of lines are **slope and intercept**.

**Slope: The Gradient of a Line**

Imagine a line traversing a hilly landscape. *Slope* quantifies the steepness of that line as it ascends or descends. It measures the **change in the vertical direction (y-axis)** divided by the corresponding **change in the horizontal direction (x-axis)**. For instance, if the line rises 3 units vertically for every 2 units it moves horizontally, the slope would be 3/2.

To calculate the slope of a line, simply divide the difference between two y-coordinates by the difference between their corresponding x-coordinates. This calculation provides a **positive value** if the line slants upward and a **negative value** if it slopes downward.

**Intercept: Where the Line Meets the Axes**

The intercept, another crucial aspect of a line, marks the **point where it crosses the y-axis**. It represents the y-coordinate of the point where the line intersects the vertical axis. The x-coordinate of this point is always zero.

Lines can also have an **x-intercept**, which indicates the point where they cross the horizontal axis. By setting the y-coordinate to zero in the equation of the line, we can determine the x-coordinate of the x-intercept.

**Slope-Intercept Form: A Convenient Representation**

The **slope-intercept form** provides a concise way to represent a line. It has the following format:

```
y = mx + b
```

Where:

**m**is the slope of the line**b**is the y-intercept**x and y**are the coordinates of any point on the line

By identifying the slope and intercept, we can fully describe the position and orientation of a line in a coordinate plane. These concepts are fundamental building blocks for understanding geometry and are indispensable for analyzing the behavior of lines and other geometric figures.

**Equation of a Line**

- Point-slope form
- Slope-intercept form

**Equation of a Line: Unveiling the Building Blocks of Geometry**

In the realm of geometry, lines play a pivotal role, defining shapes, angles, and spatial relationships. Understanding the equation of a line is essential for navigating this mathematical landscape.

**Point-Slope Form: A Simple Formula for Specific Lines**

The point-slope form of a line equation offers a straightforward method for describing lines that pass through a specific point. It follows the equation:

```
y - y1 = m(x - x1)
```

where:

- (x1, y1) is a point on the line
- m is the slope of the line

This form is particularly useful when the slope and a point on the line are known.

**Slope-Intercept Form: The Popular Choice for General Lines**

The slope-intercept form of a line equation is perhaps the most familiar form. It is the equation of a line written as:

```
y = mx + b
```

where:

- m is the slope of the line
- b is the y-intercept (the point where the line crosses the y-axis)

This form is commonly used because it clearly displays the slope and y-intercept of the line, making calculations and graphing easier.

## Parallel and Perpendicular Lines: A Geometrical Adventure

In the realm of geometry, lines intertwine, forming intricate relationships that govern their orientation and behavior. Among these relationships, the concepts of *parallel* and *perpendicular* lines stand out as fundamental building blocks.

### Parallel Lines: Side by Side, Never Intersecting

Parallel lines, like two ships sailing side by side, never meet. They share the same **slope**, which measures the steepness of a line. This means they maintain a constant vertical rise for each horizontal run, creating an illusion of parallelism that extends infinitely.

### Perpendicular Lines: Intersecting at a Right Angle

In contrast to parallel lines, perpendicular lines intersect each other at a **right angle**, forming a 90-degree angle at their point of intersection. This relationship is crucial in many real-world applications, such as architecture, engineering, and design.

### The Slope Connection

The key to understanding the relationship between parallel and perpendicular lines lies in their slopes. Parallel lines have equal slopes, while perpendicular lines have slopes that are **negative reciprocals** of each other. This means that if one line has a slope of 2, the slope of its perpendicular will be -1/2.

**Example:**

Consider two lines:

– Line 1: y = 2x + 5 (slope: 2)

– Line 2: y = -x/2 + 3 (slope: -1/2)

These lines are perpendicular because their slopes are negative reciprocals of each other (2 X (-1/2) = -1).

## Finding the Equation of a Line Perpendicular to a Given Line

Understanding perpendicular lines is crucial in geometry. When two lines **intersect** at a right angle (90 degrees), they are said to be perpendicular. **Planes** and **line segments** can also be perpendicular to each other.

To find the equation of a line perpendicular to a given line, we need to know a few things: the slope of the given line and a point on the perpendicular line.

### Determine the Slope of the Perpendicular Line

The **slope** of a line measures its **steepness**. It is calculated by dividing the **change in y** by the **change in x**. For parallel lines, their slopes are equal. However, for **perpendicular** lines, their slopes have a unique relationship: they are **negative reciprocals** of each other.

**For example**, if the given line has a slope of 2, the slope of the perpendicular line will be -1/2.

### Use the Point-Slope Form

Once we have the slope of the perpendicular line, we can use the **point-slope form** of an equation to write its equation. The point-slope form is:

```
y - y1 = m(x - x1)
```

where:

`(x1, y1)`

is a point on the line`m`

is the slope of the line

### Simplify into the Slope-Intercept Form

The point-slope form is useful for finding the equation of a line when we have a point and its slope. However, it is often more convenient to use the **slope-intercept form**, which is:

```
y = mx + b
```

where:

`m`

is the slope of the line`b`

is the y-intercept (the point where the line crosses the y-axis)

To simplify the point-slope form into the slope-intercept form, we can simply substitute `y`

for `y - y1`

and `x`

for `x - x1`

. This gives us:

```
y - y1 = m(x - x1)
y - y1 = mx - mx1
y = mx - mx1 + y1
y = mx + (y1 - mx1)
```

where `(y1 - mx1)`

is the y-intercept of the perpendicular line.

### Example

**Let’s say we have a line with the equation y = 2x + 1**. To find the equation of a line perpendicular to it, we would first calculate the slope of the given line, which is 2. The slope of the perpendicular line would then be -1/2.

**Next**, we would choose a point on the perpendicular line. Let’s say we choose the point `(1, 2)`

. Using the point-slope form, we can write the equation of the perpendicular line as:

```
y - 2 = -1/2(x - 1)
```

**Finally**, we can simplify this equation into the slope-intercept form:

```
y - 2 = -1/2x + 1/2
y = -1/2x + 3/2
```

Therefore, the equation of the line perpendicular to the given line and passing through the point `(1, 2)`

is `y = -1/2x + 3/2`

.