Understanding the Triangle Relationship

The problem provides a relationship between the sides of a triangle ABC: \(a^2 + b^2 + c^2 = ac + \sqrt{3} bc\). We are also given that the side \(c = 8\). Our goal is to find the area of this triangle.

Analyzing the Given Equation

Let's rearrange the given equation to see if it reveals any specific properties of the triangle. The equation is:

\(a^2 + b^2 + c^2 - ac - \sqrt{3} bc = 0\)

We can try to group terms and complete the square. Let's consider terms involving \(a\) and \(c\), and terms involving \(b\) and \(c\). We can rewrite the equation by splitting the \(c^2\) term.

\(a^2 - ac + \frac{c^2}{4} + b^2 - \sqrt{3} bc + \frac{3c^2}{4} = 0\)

Notice that \(\frac{c^2}{4} + \frac{3c^2}{4} = \frac{4c^2}{4} = c^2\), so the rearranged equation is equivalent to the original one.

Now, we can recognize perfect squares:

• The terms \(a^2 - ac + \frac{c^2}{4}\) form the square \(\left(a - \frac{c}{2}\right)^2\).
• The terms \(b^2 - \sqrt{3} bc + \frac{3c^2}{4}\) form the square \(\left(b - \frac{\sqrt{3}}{2}c\right)^2\).

So the equation becomes:

\(\left(a - \frac{c}{2}\right)^2 + \left(b - \frac{\sqrt{3}}{2}c\right)^2 = 0\)

Interpreting the Result

The sum of two squared terms is zero. Since squares of real numbers are always non-negative (\(\ge 0\)), this equation can only be true if both squared terms are individually zero.

• \(\left(a - \frac{c}{2}\right)^2 = 0 \implies a - \frac{c}{2} = 0 \implies a = \frac{c}{2}\)
• \(\left(b - \frac{\sqrt{3}}{2}c\right)^2 = 0 \implies b - \frac{\sqrt{3}}{2}c = 0 \implies b = \frac{\sqrt{3}}{2}c\)

This tells us the specific relationships between the sides \(a\), \(b\), and \(c\) of the triangle. The sides are in the ratio:

\(a : b : c = \frac{c}{2} : \frac{\sqrt{3}}{2}c : c = \frac{1}{2} : \frac{\sqrt{3}}{2} : 1\)

Multiplying by 2, the ratio is \(1 : \sqrt{3} : 2\). This is the characteristic ratio of sides in a 30-60-90 degree right-angled triangle.

Identifying the Triangle Type and Angles

In a triangle with sides in the ratio \(1 : \sqrt{3} : 2\), the angles opposite to these sides are \(30^\circ\), \(60^\circ\), and \(90^\circ\) respectively.

• Side \(a = c/2\) is opposite angle \(A\). Since \(a\) corresponds to the ratio 1, angle \(A = 30^\circ\).
• Side \(b = \sqrt{3}c/2\) is opposite angle \(B\). Since \(b\) corresponds to the ratio \(\sqrt{3}\), angle \(B = 60^\circ\).
• Side \(c\) is opposite angle \(C\). Since \(c\) corresponds to the ratio 2 (the hypotenuse), angle \(C = 90^\circ\).

Thus, the triangle ABC is a right-angled triangle with the right angle at C.

Calculating Side Lengths

We are given that \(c = 8\). Using the relationships we found:

• \(a = \frac{c}{2} = \frac{8}{2} = 4\)
• \(b = \frac{\sqrt{3}}{2}c = \frac{\sqrt{3}}{2}(8) = 4\sqrt{3}\)

The sides of the triangle are \(a = 4\), \(b = 4\sqrt{3}\), and \(c = 8\).

Calculating the Area of the Triangle

Since the triangle is right-angled at C, sides \(a\) and \(b\) are the legs (base and height). The area of a right-angled triangle is given by the formula:

\(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\)

Using sides \(a\) and \(b\) as the base and height:

\(\text{Area} = \frac{1}{2}ab\)

Substitute the values of \(a\) and \(b\):

\(\text{Area} = \frac{1}{2} \times 4 \times 4\sqrt{3}\)

\(\text{Area} = \frac{1}{2} \times 16\sqrt{3}\)

\(\text{Area} = 8\sqrt{3}\)

Alternatively, using the general area formula \(\text{Area} = \frac{1}{2}ab \sin C\):

With \(a = 4\), \(b = 4\sqrt{3}\), and \(C = 90^\circ\), we have \(\sin 90^\circ = 1\).

\(\text{Area} = \frac{1}{2} \times 4 \times 4\sqrt{3} \times \sin 90^\circ\)

\(\text{Area} = \frac{1}{2} \times 16\sqrt{3} \times 1\)

\(\text{Area} = 8\sqrt{3}\)

Conclusion

The area of the triangle with the given properties and \(c=8\) is \(8\sqrt{3}\).

Comparing this result with the given options:

• Option 1: \(4\sqrt{3}\)
• Option 2: \(6\sqrt{3}\)
• Option 3: \(8\sqrt{3}\)
• Option 4: \(12\sqrt{3}\)

The calculated area matches Option 3.

Revision Table: Triangle Area Calculation

Additional Information: Triangle Properties and Formulas

Understanding relationships between sides and angles is fundamental in solving triangle problems. Here are some key concepts:

• Law of Cosines: Relates the lengths of sides of a triangle to the cosine of one of its angles. The formulas are:
  • \(a^2 = b^2 + c^2 - 2bc \cos A\)
  • \(b^2 = a^2 + c^2 - 2ac \cos B\)
  • \(c^2 = a^2 + b^2 - 2ab \cos C\)
• Law of Sines: Relates the ratio of the length of a side to the sine of its opposite angle. The formula is:

  \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R\)

  where \(R\) is the circumradius of the triangle.

• Area Formulas:
  • Base and Height: \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\)
  • Two Sides and Included Angle: \(\text{Area} = \frac{1}{2}ab \sin C = \frac{1}{2}bc \sin A = \frac{1}{2}ac \sin B\)
  • Heron's Formula (using semi-perimeter \(s = (a+b+c)/2\)): \(\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}\)
• 30-60-90 Triangle: A right triangle whose angles are \(30^\circ\), \(60^\circ\), and \(90^\circ\). The sides opposite these angles are in the ratio \(1 : \sqrt{3} : 2\). This special triangle frequently appears in geometry and trigonometry problems.

Recognizing specific side relationships or angle measures can significantly simplify solving triangle problems.