This worksheet features a variety of triangle problems designed to test skills in Pythagoras' theorem and basic trigonometry.

To solve the problems in the image, we will use Pythagoras' Theorem and Trigonometric Ratios as required. Let's go through each problem step by step.

---

Top Row:

#### 1.
Given:
- Right triangle with legs 23 cm and 26 cm.
- Hypotenuse = \( a \).

Using Pythagoras' Theorem:
\[
a^2 = 23^2 + 26^2
\]
\[
a^2 = 529 + 676
\]
\[
a^2 = 1205
\]
\[
a = \sqrt{1205} \approx 34.71 \text{ cm}
\]

#### 2.
Given:
- Right triangle with one leg = 25 cm, hypotenuse = 40 cm.
- Other leg = \( b \).

Using Pythagoras' Theorem:
\[
40^2 = 25^2 + b^2
\]
\[
1600 = 625 + b^2
\]
\[
b^2 = 1600 - 625
\]
\[
b^2 = 975
\]
\[
b = \sqrt{975} \approx 31.22 \text{ cm}
\]

#### 3.
Given:
- Right triangle with one leg = 52 cm, angle = 41°.
- Hypotenuse = \( c \).

Using trigonometric ratios (cosine):
\[
\cos(41^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{52}{c}
\]
\[
c = \frac{52}{\cos(41^\circ)}
\]
\[
c \approx \frac{52}{0.7547} \approx 68.91 \text{ cm}
\]

#### 4.
Given:
- Right triangle with one leg = 44 cm, angle = 29°.
- Hypotenuse = \( d \).

Using trigonometric ratios (sine):
\[
\sin(29^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{44}{d}
\]
\[
d = \frac{44}{\sin(29^\circ)}
\]
\[
d \approx \frac{44}{0.4848} \approx 90.74 \text{ cm}
\]

---

Middle Row:

#### 5.
Given:
- Right triangle with one leg = 27 cm, angle = 56°.
- Hypotenuse = \( f \).

Using trigonometric ratios (cosine):
\[
\cos(56^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{27}{f}
\]
\[
f = \frac{27}{\cos(56^\circ)}
\]
\[
f \approx \frac{27}{0.5592} \approx 48.49 \text{ cm}
\]

#### 6.
Given:
- Right triangle with one leg = 6 cm, hypotenuse = 27 cm.
- Other leg = \( g \).

Using Pythagoras' Theorem:
\[
27^2 = 6^2 + g^2
\]
\[
729 = 36 + g^2
\]
\[
g^2 = 729 - 36
\]
\[
g^2 = 693
\]
\[
g = \sqrt{693} \approx 26.32 \text{ cm}
\]

#### 7.
Given:
- Right triangle with one leg = 18 cm, other leg = 6 cm.
- Hypotenuse = \( h \).

Using Pythagoras' Theorem:
\[
h^2 = 18^2 + 6^2
\]
\[
h^2 = 324 + 36
\]
\[
h^2 = 360
\]
\[
h = \sqrt{360} \approx 18.97 \text{ cm}
\]

#### 8.
Given:
- Right triangle with one leg = 48 cm, hypotenuse = 36 cm.
- Angle = 44°.
- Other leg = \( i \).

Using trigonometric ratios (tangent):
\[
\tan(44^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{i}{48}
\]
\[
i = 48 \cdot \tan(44^\circ)
\]
\[
i \approx 48 \cdot 0.9657 \approx 46.35 \text{ cm}
\]

---

Bottom Row:

#### 9.
Given:
- Right triangle with one leg = 35 cm, angle = 36°.
- Hypotenuse = \( k \).

Using trigonometric ratios (sine):
\[
\sin(36^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{35}{k}
\]
\[
k = \frac{35}{\sin(36^\circ)}
\]
\[
k \approx \frac{35}{0.5878} \approx 59.52 \text{ cm}
\]

#### 10.
Given:
- Right triangle with one leg = 26 cm, angle = 41°.
- Hypotenuse = \( l \).

Using trigonometric ratios (cosine):
\[
\cos(41^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{26}{l}
\]
\[
l = \frac{26}{\cos(41^\circ)}
\]
\[
l \approx \frac{26}{0.7547} \approx 34.45 \text{ cm}
\]

#### 11.
Given:
- Triangle with sides 47 cm, 71 cm, and an included angle of 71°.
- We need to find the third side (\( m \)) using the Law of Cosines (not Pythagoras or trigonometric ratios directly).

Using the Law of Cosines:
\[
m^2 = 47^2 + 71^2 - 2 \cdot 47 \cdot 71 \cdot \cos(71^\circ)
\]
\[
m^2 = 2209 + 5041 - 2 \cdot 47 \cdot 71 \cdot 0.3256
\]
\[
m^2 = 2209 + 5041 - 2209 \cdot 0.3256
\]
\[
m^2 = 2209 + 5041 - 719.97
\]
\[
m^2 = 6530.03
\]
\[
m \approx \sqrt{6530.03} \approx 80.81 \text{ cm}
\]

#### 12.
Given:
- Trapezoid with bases 7 cm and 11 cm, height = 78°.
- We need to find the slant height (\( n \)).

Using trigonometric ratios (tangent):
\[
\tan(78^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{11 - 7}{n}
\]
\[
\tan(78^\circ) = \frac{4}{n}
\]
\[
n = \frac{4}{\tan(78^\circ)}
\]
\[
n \approx \frac{4}{4.7046} \approx 0.85 \text{ cm}
\]

---

Final Answers:

\[
\boxed{
\begin{array}{cccc}
a \approx 34.71 & b \approx 31.22 & c \approx 68.91 & d \approx 90.74 \\
f \approx 48.49 & g \approx 26.32 & h \approx 18.97 & i \approx 46.35 \\
k \approx 59.52 & l \approx 34.45 & m \approx 80.81 & n \approx 0.85 \\
\end{array}
}
\]