https://file.aiquickdraw.com/mathbot/file/1730633411798vdw5za6p.jpg

Let's analyze and solve each part of this problem step-by-step.

### Problem Breakdown
1. We are given a square with sides parallel to the x- and y-axes.
2. The center of the square is at the origin, \((0, 0)\).
3. Each side of the square is 8 units long.

### Part (a)
**Objective**: Find the equations of each of the four lines that form the sides of the square.

#### Solution:
Since the center is at \((0, 0)\) and the side length is 8 units, each side extends 4 units from the center in both x and y directions.

1. **Lines parallel to the x-axis**:
   - The top side is 4 units above the center: \(y = 4\).
   - The bottom side is 4 units below the center: \(y = -4\).

2. **Lines parallel to the y-axis**:
   - The right side is 4 units to the right of the center: \(x = 4\).
   - The left side is 4 units to the left of the center: \(x = -4\).

Therefore, the equations of the four lines are:
   - \(y = 4\)
   - \(y = -4\)
   - \(x = 4\)
   - \(x = -4\)

---

### Part (b)
**Objective**: A second square is the same size but its center is in a different place. How would you find the equations of the four lines of this square?

#### Solution:
If the new center is at \((h, k)\), then each side will again be 4 units away from the center along both the x and y directions.

1. **Lines parallel to the x-axis**:
   - The top side: \(y = k + 4\)
   - The bottom side: \(y = k - 4\)

2. **Lines parallel to the y-axis**:
   - The right side: \(x = h + 4\)
   - The left side: \(x = h - 4\)

So, the equations of the lines for the square centered at \((h, k)\) are:
   - \(y = k + 4\)
   - \(y = k - 4\)
   - \(x = h + 4\)
   - \(x = h - 4\)

---

### Part (c)
**Objective**: Suppose you know the center of a square and the length of each side. How would you find the equations of the four lines?

#### Solution:
Let’s assume the center of the square is \((h, k)\) and the side length is \(s\).

1. Since the side length is \(s\), the distance from the center to each side will be \(s/2\).

2. **Lines parallel to the x-axis**:
   - The top side: \(y = k + s/2\)
   - The bottom side: \(y = k - s/2\)

3. **Lines parallel to the y-axis**:
   - The right side: \(x = h + s/2\)
   - The left side: \(x = h - s/2\)

Thus, for a square centered at \((h, k)\) with side length \(s\), the equations are:
   - \(y = k + s/2\)
   - \(y = k - s/2\)
   - \(x = h + s/2\)
   - \(x = h - s/2\)

---

Would you like further explanation on any part of these solutions? 

### Related Questions:
1. How would the equations change if the square were rotated by 45 degrees?
2. How can you verify that these equations indeed form a square?
3. How would you find the area of the square if you only had the equations of the lines?
4. If you know the equations of two opposite sides of the square, how can you determine the center?
5. How would the equations of the sides change if the square were centered at a point on the line \(y = x\)?

### Tip:
When dealing with squares or rectangles centered on the origin, symmetry can often simplify calculations.

Two lines parallel to the x-axis and two lines parallel to the y-axis make a square. The centre of the square is at the origin (0, 0). Each side of the square is 8 units.

(a) Find the equation of each of the four lines.
(b) A second square is the same size but its centre is in a different place. How would you find the equations of the four lines of this square?
(c) Suppose you know the centre of a square and the length of each side. How would you find the equations of the four lines?

Learn how to find the equations of lines forming a square centered at the origin with side length 8 units. Explore additional concepts for squares centered at any point and how to derive equations based on given center and side length. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation