Find the length of the indicated side, to the nearest tenth of a metre.

Steps to Solve

1. Identify the triangle components
   In triangle ( RKT ), we have two sides and an angle:

• Side ( RT = 4.3 , \text{m} )
• Side ( RK = 3.6 , \text{m} )
• Angle ( K = 36^\circ )

2. Use the Law of Cosines
   To find the side labeled ( X ) (which is ( KT )), we will use the Law of Cosines formula:
   $$ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) $$
   where:

• ( c ) is the side opposite the angle ( C ) (in this case, ( KT = X ))
• ( a ) and ( b ) are the other two sides (here, ( RT ) and ( RK ))
• ( C ) is the angle ( K )

Thus, our equation becomes:
$$ X^2 = (4.3)^2 + (3.6)^2 - 2 \cdot (4.3) \cdot (3.6) \cdot \cos(36^\circ) $$

3. Calculate the squares of the sides
   Calculate ( (4.3)^2 ) and ( (3.6)^2 ):
   $$ (4.3)^2 = 18.49 $$
   $$ (3.6)^2 = 12.96 $$

4. Calculate the cosine component
   Find ( \cos(36^\circ) ) approximately:
   $$ \cos(36^\circ) \approx 0.8090 $$

Now substitute values into the equation:
$$ X^2 = 18.49 + 12.96 - 2 \cdot 4.3 \cdot 3.6 \cdot 0.8090 $$

5. Calculate the full expression
   Now we can compute the value:
   $$ 2 \cdot 4.3 \cdot 3.6 \cdot 0.8090 \approx 26.517 $$
   So,
   $$ X^2 = 18.49 + 12.96 - 26.517 $$
   $$ X^2 = 31.45 - 26.517 = 4.933 $$

6. Take the square root to find X
   Finally, take the square root:
   $$ X = \sqrt{4.933} \approx 2.22 , \text{m} $$

Rounding to the nearest tenth:
$$ X \approx 2.2 , \text{m} $$