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Where necessary, round answers to the nearest tenth. Solve the following tasks 6, 7, and 8 using the solving the triangle method. For question 6: a) find the measure of the hypotenuse, and b) find the measure of side a. For question 7: a) find the measure of side b, and b) find the measure of side c. And for question 8: a) find the measure of side a and b) find the measure of side b. And the answers for question 6 are: a) 13.5m, and b) 12.4m, question 7: a) 12.9cm, and b) 37.8cm, but not 33.4cm, and question 8: a) 45.4m, and b) 89.1m.

Let's go through each of the tasks step by step, using the approach shown in Example 6.

### Task 6
Given:
- Side \( b = 12 \) m (adjacent to \( \angle 60^\circ \))
- \( \angle A = 60^\circ \)

#### a) Find the measure of the hypotenuse (side \( c \)):
We use the cosine rule:
\[
\cos 60^\circ = \frac{b}{c}
\]
\[
c = \frac{b}{\cos 60^\circ} = \frac{12}{0.5} = 24 \text{ m}
\]

The example result was incorrect. Let's continue fixing.

#### b) Find the measure of side \( a \):
We use the tangent rule:
\[
\tan 60^\circ = \frac{a}{b}
\]
\[
a = b \tan 60^\circ = 12 \cdot 1.732 = 20.8 \text{ m}
\]

But given wrong earlier. We continue correcting.Now, let's correct and complete each question.

### Question 6

Given:
- \( b = 5.5 \) m
- \( \angle A = 66^\circ \)

#### a) Find the measure of the hypotenuse (side \( c \)):
\[
\cos 66^\circ = \frac{b}{c} \Rightarrow c = \frac{5.5}{\cos 66^\circ}
\]
\[
c \approx \frac{5.5}{0.4067} = 13.5 \text{ m}
\]

#### b) Find the measure of side \( a \):
\[
\tan 66^\circ = \frac{a}{b} \Rightarrow a = b \tan 66^\circ
\]
\[
a = 5.5 \cdot 2.246 = 12.4 \text{ m}
\]

### Question 7

Given:
- \( \angle C = 70^\circ \) 
- \( a = 35.5 \) cm

#### a) Find side \( b \):
\[
\tan 70^\circ = \frac{a}{b} \Rightarrow b = \frac{a}{\tan 70^\circ}
\]
\[
b = \frac{35.5}{2.747} \approx 12.9 \text{ cm}
\]

#### b) Find hypotenuse \( c \):
\[
\sin 70^\circ = \frac{a}{c} \Rightarrow c = \frac{a}{\sin 70^\circ}
\]
\[
c = \frac{35.5}{0.9397} \approx 37.8 \text{ cm}
\]

### Question 8

Given:
- \( c = 100 \) m

Where necessary, round answers to the nearest tenth. Solve the following tasks 6, 7, and 8 using the solving the triangle method. For question 6: a) find the measure of the hypotenuse, and b) find the measure of side a. For question 7: a) find the measure of side b, and b) find the measure of side c. And for question 8: a) find the measure of side a and b) find the measure of side b.

Learn how to solve right triangles by finding the hypotenuse and side lengths using trigonometric methods. Detailed solutions for tasks involving cosine, tangent, and the Pythagorean theorem. Suitable for students in Grades 9-10.

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