Mixed Pythagoras & Trigonometry Questions - Go Teach Maths ... - Free Printable

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:

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