1. Construct a square $$ A B C D $$, when a diagonal $$ \overline{A C}=4.5 \mathrm{~cm} $$. Construct a square PQRS whose diagonal is 4 cm more than its side. Construct a square PORS, when the sum of the diagonal and a side of the square is known.

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Sure! Let's address each of your construction problems step-by-step. We'll use fundamental geometric principles and algebraic relationships pertinent to squares.

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## 1. **Constructing a Square $$ ABCD $$ with Diagonal $$ \overline{AC} = 4.5 $$ cm**

### **Understanding the Relationship:**
In a square:
- All sides are equal.
- The diagonals are equal and bisect each other at 90°, intersecting at the square's center.
- The diagonal ($$ d $$) relates to the side length ($$ a $$) by the formula:
  $$
  d = a \sqrt{2}
  $$
  
### **Steps for Construction:**

1. **Draw Diagonal $$ AC $$:**
   - Use a ruler to draw a straight line segment $$ AC $$ of length **4.5 cm**.

2. **Find the Midpoint $$ O $$ of $$ AC $$:**
   - Use a ruler to measure the length of $$ AC $$ and mark its midpoint as point $$ O $$.

3. **Construct a Perpendicular Line at $$ O $$:**
   - Using a compass and straightedge, draw a line perpendicular to $$ AC $$ at point $$ O $$. This will be the other diagonal $$ BD $$.

4. **Determine the Length of Half the Diagonal $$ BD $$:**
   - Since the full diagonal of a square is $$ d = a \sqrt{2} $$, half of the diagonal is $$ \frac{d}{2} = \frac{4.5}{2} = 2.25 $$ cm.
   - However, due to the perpendicular intersection, the length from $$ O $$ to $$ B $$ (and $$ O $$ to $$ D $$) should also satisfy $$ \frac{d}{2} = \frac{a \sqrt{2}}{2} $$. Given that $$ d = 4.5 $$ cm, we can find:
     $$
     a = \frac{d}{\sqrt{2}} = \frac{4.5}{\sqrt{2}} \approx 3.181 \text{ cm}
     $$
   
5. **Mark Points $$ B $$ and $$ D $$:**
   - With the compass set to approximately **3.181 cm**, place the compass point at $$ O $$ and mark points $$ B $$ and $$ D $$ on the perpendicular line.

6. **Complete the Square:**
   - Connect points $$ A $$ to $$ B $$, $$ B $$ to $$ C $$, $$ C $$ to $$ D $$, and $$ D $$ back to $$ A $$ using a ruler to form square $$ ABCD $$.

### **Result:**
You will have constructed square $$ ABCD $$ with diagonals intersecting at 4.5 cm, ensuring all sides are equal in length.

---

## 2. **Constructing a Square $$ PQRS $$ Whose Diagonal is 4 cm More Than Its Side**

### **Understanding the Relationship:**
Let:
- $$ a $$ = side length of the square.
- $$ d $$ = diagonal of the square.

Given:
$$
d = a + 4 \quad \text{cm}
$$
And from the square's properties:
$$
d = a \sqrt{2}
$$

### **Solving for the Side Length $$ a $$:**
Set the two expressions for $$ d $$ equal to each other:
$$
a \sqrt{2} = a + 4
$$
$$
a \sqrt{2} - a = 4
$$
$$
a (\sqrt{2} - 1) = 4
$$
$$
a = \frac{4}{\sqrt{2} - 1}
$$
Rationalize the denominator:
$$
a = \frac{4(\sqrt{2} + 1)}{(\sqrt{2} - 1)(\sqrt{2} + 1)} = \frac{4(\sqrt{2} + 1)}{2 - 1} = 4(\sqrt{2} + 1) \quad \text{cm}
$$
$$
a \approx 4(1.4142 + 1) = 4(2.4142) \approx 9.656 \quad \text{cm}
$$

### **Steps for Construction:**

1. **Calculate Side Length $$ a $$:**
   - As derived, $$ a = 4(\sqrt{2} + 1) $$ cm. For practical purposes, you can use the approximate value of **9.656 cm**.

2. **Draw Side $$ PQ $$:**
   - Use a ruler to draw a straight line segment $$ PQ $$ of length **9.656 cm**.

3. **Construct Perpendicular Lines for Remaining Sides:**
   - At point $$ P $$, use a protractor or compass to draw a line perpendicular to $$ PQ $$.
   - Similarly, at point $$ Q $$, draw a line perpendicular to $$ PQ $$.

4. **Mark Points $$ R $$ and $$ S $$:**
   - Measure **9.656 cm** along the perpendicular lines from $$ P $$ and $$ Q $$ to locate points $$ S $$ and $$ R $$, respectively.

5. **Connect the Points:**
   - Draw lines $$ QR $$ and $$ SR $$ to complete square $$ PQRS $$.

6. **Verify the Diagonal:**
   - Measure the diagonal $$ PR $$ to ensure it is approximately **13.656 cm** (since $$ d = a + 4 = 9.656 + 4 = 13.656 $$ cm).

### **Result:**
You've successfully constructed square $$ PQRS $$ where the diagonal is 4 cm longer than its side.

---

## 3. **Constructing a Square $$ PORS $$ When the Sum of the Diagonal and a Side is Known**

### **Understanding the Relationship:**
Let:
- $$ a $$ = side length of the square.
- $$ d $$ = diagonal of the square.
- $$ c $$ = known sum of the diagonal and the side.

Given:
$$
d + a = c
$$
And from the square's properties:
$$
d = a \sqrt{2}
$$

### **Solving for the Side Length $$ a $$:**
Substitute $$ d $$ from the second equation into the first:
$$
a \sqrt{2} + a = c
$$
$$
a (\sqrt{2} + 1) = c
$$
$$
a = \frac{c}{\sqrt{2} + 1}
$$
Rationalize the denominator:
$$
a = \frac{c (\sqrt{2} - 1)}{(\sqrt{2} + 1)(\sqrt{2} - 1)} = \frac{c (\sqrt{2} - 1)}{2 - 1} = c (\sqrt{2} - 1) \quad \text{cm}
$$
$$
a \approx c (1.4142 - 1) = c (0.4142) \quad \text{cm}
$$

### **Steps for Construction:**

1. **Calculate Side Length $$ a $$:**
   - Based on the known sum $$ c $$, compute $$ a = c (\sqrt{2} - 1) $$. For example, if $$ c = 10 $$ cm:
     $$
     a \approx 10 \times 0.4142 = 4.142 \quad \text{cm}
     $$

2. **Draw Side $$ PO $$:**
   - Use a ruler to draw a straight line segment $$ PO $$ of length $$ a $$ cm (e.g., **4.142 cm**).

3. **Construct Perpendicular Lines for Remaining Sides:**
   - At point $$ P $$, use a protractor or compass to draw a line perpendicular to $$ PO $$.
   - Similarly, at point $$ O $$, draw a line perpendicular to $$ PO $$.

4. **Mark Points $$ R $$ and $$ S $$:**
   - Measure the length $$ a $$ along the perpendicular lines from $$ P $$ and $$ O $$ to locate points $$ S $$ and $$ R $$, respectively.

5. **Connect the Points:**
   - Draw lines $$ OR $$, $$ RS $$, and $$ SP $$ to complete square $$ PORS $$.

6. **Verify the Sum $$ d + a $$:**
   - Measure the diagonal $$ PR $$ to ensure that $$ d + a = c $$. Given $$ a $$, $$ d = a \sqrt{2} $$, so:
     $$
     d + a = a (\sqrt{2} + 1) = c
     $$

### **Result:**
You've successfully constructed square $$ PORS $$ where the sum of the diagonal and the side equals the known value $$ c $$.

---

These constructions rely on understanding the fundamental properties of squares and applying algebraic relationships to determine necessary lengths. By following these steps, you can accurately construct each square based on the given conditions.
To construct a square with a diagonal of 4.5 cm, draw the diagonal and find its midpoint. Then, draw a perpendicular line at the midpoint and mark points to complete the square. For a square where the diagonal is 4 cm more than its side, calculate the side length using the formula $$ a = \frac{4}{\sqrt{2} - 1} $$ cm, then draw the square accordingly. When the sum of the diagonal and a side is known, use the formula $$ a = c (\sqrt{2} - 1) $$ cm to find the side length and construct the square.

Construct a square , when a diagonal .

Construct a square PQRS whose diagonal is 4 cm more than its side.
Construct a square PORS, when the sum of the diagonal and a side of the square is known.

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Construct a square , when a diagonal . Construct a square PQRS whose diagonal is 4 cm more than its side. Construct a square PORS, when the sum of the diagonal and a side of the square is known.