Find the perimeter and area of the given geometric shapes.

To solve the problem, we need to calculate the perimeter and area for each of the given shapes. Let's go through each shape step by step.

---

Shape 1: Rectangle with a cut-out

#### Dimensions:
- Outer rectangle: \(76 \, \text{cm} \times 38 \, \text{cm}\)
- Inner rectangle (cut-out): \(24 \, \text{cm} \times 14 \, \text{cm}\)

#### Perimeter:
The perimeter of the outer rectangle is:
\[
P_{\text{outer}} = 2 \times (76 + 38) = 2 \times 114 = 228 \, \text{cm}
\]
The inner rectangle does not affect the perimeter since it is a cut-out. Therefore, the perimeter remains:
\[
\boxed{228 \, \text{cm}}
\]

#### Area:
The area of the outer rectangle is:
\[
A_{\text{outer}} = 76 \times 38 = 2888 \, \text{cm}^2
\]
The area of the inner rectangle is:
\[
A_{\text{inner}} = 24 \times 14 = 336 \, \text{cm}^2
\]
The area of the shape is the area of the outer rectangle minus the area of the inner rectangle:
\[
A = 2888 - 336 = 2552 \, \text{cm}^2
\]
\[
\boxed{2552 \, \text{cm}^2}
\]

---

Shape 2: Triangle

#### Dimensions:
- Base: \(15 \, \text{cm}\)
- Height: \(58 \, \text{cm}\)
- Hypotenuse: \(29 \, \text{cm}\)

#### Perimeter:
The perimeter is the sum of all sides:
\[
P = 15 + 58 + 29 = 102 \, \text{cm}
\]
\[
\boxed{102 \, \text{cm}}
\]

#### Area:
The area of a triangle is given by:
\[
A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 15 \times 58 = \frac{1}{2} \times 870 = 435 \, \text{cm}^2
\]
\[
\boxed{435 \, \text{cm}^2}
\]

---

Shape 3: Triangle

#### Dimensions:
- Sides: \(52 \, \text{cm}\), \(58 \, \text{cm}\), \(29 \, \text{cm}\)

#### Perimeter:
The perimeter is the sum of all sides:
\[
P = 52 + 58 + 29 = 139 \, \text{cm}
\]
\[
\boxed{139 \, \text{cm}}
\]

#### Area:
We use Heron's formula to find the area. First, calculate the semi-perimeter \(s\):
\[
s = \frac{52 + 58 + 29}{2} = \frac{139}{2} = 69.5 \, \text{cm}
\]
Next, use Heron's formula:
\[
A = \sqrt{s(s-a)(s-b)(s-c)}
\]
where \(a = 52\), \(b = 58\), and \(c = 29\):
\[
A = \sqrt{69.5 \times (69.5 - 52) \times (69.5 - 58) \times (69.5 - 29)}
\]
\[
A = \sqrt{69.5 \times 17.5 \times 11.5 \times 40.5}
\]
Calculate step-by-step:
\[
69.5 \times 17.5 = 1226.25
\]
\[
11.5 \times 40.5 = 465.75
\]
\[
1226.25 \times 465.75 = 572,109.375
\]
\[
A = \sqrt{572,109.375} \approx 756.38 \, \text{cm}^2
\]
\[
\boxed{756.38 \, \text{cm}^2}
\]

---

Shape 4: Rectangle

#### Dimensions:
- Length: \(78 \, \text{cm}\)
- Width: \(68 \, \text{cm}\)

#### Perimeter:
The perimeter of a rectangle is:
\[
P = 2 \times (\text{length} + \text{width}) = 2 \times (78 + 68) = 2 \times 146 = 292 \, \text{cm}
\]
\[
\boxed{292 \, \text{cm}}
\]

#### Area:
The area of a rectangle is:
\[
A = \text{length} \times \text{width} = 78 \times 68 = 5304 \, \text{cm}^2
\]
\[
\boxed{5304 \, \text{cm}^2}
\]

---

Shape 5: Isosceles Triangle

#### Dimensions:
- Base: \(31 \, \text{cm}\)
- Equal sides: \(31 \, \text{cm}\)
- Height: \(42 \, \text{cm}\)

#### Perimeter:
The perimeter is the sum of all sides:
\[
P = 31 + 31 + 31 = 93 \, \text{cm}
\]
\[
\boxed{93 \, \text{cm}}
\]

#### Area:
The area of a triangle is:
\[
A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 31 \times 42 = \frac{1}{2} \times 1302 = 651 \, \text{cm}^2
\]
\[
\boxed{651 \, \text{cm}^2}
\]

---

Shape 6: Right Triangle

#### Dimensions:
- Base: \(25 \, \text{cm}\)
- Height: \(36 \, \text{cm}\)
- Hypotenuse: \(40 \, \text{cm}\)

#### Perimeter:
The perimeter is the sum of all sides:
\[
P = 25 + 36 + 40 = 101 \, \text{cm}
\]
\[
\boxed{101 \, \text{cm}}
\]

#### Area:
The area of a triangle is:
\[
A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 25 \times 36 = \frac{1}{2} \times 900 = 450 \, \text{cm}^2
\]
\[
\boxed{450 \, \text{cm}^2}
\]

---

Final Answers:

1. Rectangle with cut-out:
- Perimeter: \(228 \, \text{cm}\)
- Area: \(2552 \, \text{cm}^2\)

2. Triangle:
- Perimeter: \(102 \, \text{cm}\)
- Area: \(435 \, \text{cm}^2\)

3. Triangle:
- Perimeter: \(139 \, \text{cm}\)
- Area: \(756.38 \, \text{cm}^2\)

4. Rectangle:
- Perimeter: \(292 \, \text{cm}\)
- Area: \(5304 \, \text{cm}^2\)

5. Isosceles Triangle:
- Perimeter: \(93 \, \text{cm}\)
- Area: \(651 \, \text{cm}^2\)

6. Right Triangle:
- Perimeter: \(101 \, \text{cm}\)
- Area: \(450 \, \text{cm}^2\)

\[
\boxed{
\begin{array}{ll}
\text{1. Perimeter: } 228 \, \text{cm}, & \text{Area: } 2552 \, \text{cm}^2 \\
\text{2. Perimeter: } 102 \, \text{cm}, & \text{Area: } 435 \, \text{cm}^2 \\
\text{3. Perimeter: } 139 \, \text{cm}, & \text{Area: } 756.38 \, \text{cm}^2 \\
\text{4. Perimeter: } 292 \, \text{cm}, & \text{Area: } 5304 \, \text{cm}^2 \\
\text{5. Perimeter: } 93 \, \text{cm}, & \text{Area: } 651 \, \text{cm}^2 \\
\text{6. Perimeter: } 101 \, \text{cm}, & \text{Area: } 450 \, \text{cm}^2 \\
\end{array}
}
\]