Given the right triangle $ABC$ with:

• Right angle at $C$,
• Angle $B = 72^\circ$,
• Side $b = 6.8$ (the length of side $BC$),

We are asked to find the remaining sides and angles:

Step 1: Determine angle $A$

Since the sum of all angles in a triangle is $180^\circ$, and there is a right angle at $C$, we can find angle $A$ as follows:

$A = 180^\circ - 90^\circ - 72^\circ = 18^\circ$

Step 2: Use the Law of Sines to find side $a$ (length of side $AC$)

The Law of Sines states:

$\frac{a}{\sin A} = \frac{b}{\sin B}$

Substituting the known values:

$\frac{a}{\sin 18^\circ} = \frac{6.8}{\sin 72^\circ}$

Now, we solve for $a$:

$a = \frac{6.8 \cdot \sin 18^\circ}{\sin 72^\circ}$

Using the approximate values $\sin 18^\circ \approx 0.309$ and $\sin 72^\circ \approx 0.951$:

$a = \frac{6.8 \cdot 0.309}{0.951} \approx \frac{2.103}{0.951} \approx 2.21$

So, the length of side $a$ is approximately $2.2$.

Step 3: Use the Pythagorean Theorem to find side $c$ (length of side $AB$)

The Pythagorean Theorem states:

$c^2 = a^2 + b^2$

Substitute the known values for $a$ and $b$:

$c^2 = (2.2)^2 + (6.8)^2 = 4.84 + 46.24 = 51.08$

Taking the square root of both sides:

$c \approx \sqrt{51.08} \approx 7.15$

Final Answer:

• Angle $A = 18^\circ$
• Side $a \approx 2.2$
• Side $c \approx 7.15$

Would you like a further explanation of any part, or is there anything else you'd like to explore?

Here are some follow-up questions:

1. How do you apply the Law of Sines in a non-right triangle?
2. What are other methods to find missing sides or angles in right triangles?
3. How can you verify your results using different trigonometric identities?
4. What role does the Pythagorean Theorem play in determining the sides of a triangle?
5. How would you approach a triangle with no right angle (oblique triangle)?

Tip: Always double-check your angle sum in a triangle to avoid common mistakes!