EVALUATE:

Solve the following problems accurately. Show your complete solution.

Solutions must be handwritten. (10 pts each)

1.    Find the area of a triangle whose sides are 16 cm, 18 cm and 28 cm.



2.    The perimeter of a rectangle is 80 cm. If the length is 14 cm. Find its width and area. How long is the diagonal?

 

3.    The diagonal of a square is 32 cm. Find the area, the side, and the perimeter of a square.

Sure! Here are the solutions to the problems you provided, including complete steps for each calculation.

### Problem 1: Area of a Triangle

To find the area of a triangle given the sides, we can use **Heron's formula**.

1. **Calculate the semi-perimeter \( s \)**:
   \[
   s = \frac{a + b + c}{2} = \frac{16 + 18 + 28}{2} = 31 \text{ cm}
   \]

2. **Calculate the area \( A \)** using Heron's formula:
   \[
   A = \sqrt{s(s-a)(s-b)(s-c)}
   \]
   where \( a = 16 \), \( b = 18 \), and \( c = 28 \):
   \[
   A = \sqrt{31(31-16)(31-18)(31-28)} = \sqrt{31 \times 15 \times 13 \times 3}
   \]
   \[
   A = \sqrt{31 \times 15 \times 39}
   \]
   \[
   A = \sqrt{31 \times 585} \approx 140.5 \text{ cm}^2
   \]

### Problem 2: Rectangle Perimeter, Width, Area, and Diagonal

Given the perimeter \( P \) of the rectangle and its length \( l \):
1. **Use the perimeter formula**:
   \[
   P = 2(l + w) \Rightarrow 80 = 2(14 + w)
   \]
   \[
   40 = 14 + w \Rightarrow w = 40 - 14 = 26 \text{ cm}
   \]

2. **Calculate the area \( A \)**:
   \[
   A = l \times w = 14 \times 26 = 364 \text{ cm}^2
   \]

3. **Calculate the diagonal \( d \)** using the Pythagorean theorem:
   \[
   d = \sqrt{l^2 + w^2} = \sqrt{14^2 + 26^2} = \sqrt{196 + 676} = \sqrt{872} \approx 29.5 \text{ cm}
   \]

### Problem 3: Square Diagonal, Area, Side, and Perimeter

Given the diagonal \( d \) of the square:
1. **Calculate the side length \( s \)**:
   \[
   d = s\sqrt{2} \Rightarrow s = \frac{d}{\sqrt{2}} = \frac{32}{\sqrt{2}} = 16\sqrt{2} \text{ cm}
   \]

2. **Calculate the area \( A \)**:
   \[
   A = s^2 = (16\sqrt{2})^2 = 256 \times 2 = 512 \text{ cm}^2
   \]

3. **Calculate the perimeter \( P \)**:
   \[
   P = 4s = 4 \times 16\sqrt{2} \approx 64\sqrt{2} \approx 90.5 \text{ cm}
   \]

### Summary of Solutions
1. Area of the triangle: **approximately 140.5 cm²**
2. Width of the rectangle: **26 cm**, Area: **364 cm²**, Diagonal: **approximately 29.5 cm**
3. Area of the square: **512 cm²**, Side: **approximately 22.63 cm**, Perimeter: **approximately 90.5 cm**

Do you want details on any specific part or have any questions? Here are some related questions to consider:

1. How does Heron's formula work for triangles with different side lengths?
2. What is the relationship between the diagonal and the sides of a rectangle?
3. How does the area of a square change if the diagonal is doubled?
4. What other methods can be used to calculate the area of a triangle?
5. Can you explain the Pythagorean theorem in more detail?

**Tip**: Always double-check your calculations for accuracy, especially in geometric problems!

Find the area of a triangle whose sides are 16 cm, 18 cm and 28 cm. The perimeter of a rectangle is 80 cm. If the length is 14 cm, find its width and area. How long is the diagonal? The diagonal of a square is 32 cm. Find the area, the side, and the perimeter of a square.

Explore solutions for finding the area of a triangle with sides 16 cm, 18 cm, and 28 cm using Heron's formula, calculating the width and area of a rectangle with a perimeter of 80 cm and a length of 14 cm, and determining the area, side, and perimeter of a square with a diagonal of 32 cm. Perfect for students in Grades 7-10.

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