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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!