Question

The area of the triangle ABC is $10\sqrt{3}\ \text{cm}^2$, angle B is $60^\circ$ and its perimeter is $20\ \text{cm}$, then $\ell(\text{AC}) =$

• A. 10 cm
• B. 8 cm
• C. 5 cm
• D. 7 cm

Hint:

Whenever you see a system of equations where you have values for both $ac$ and $a+c$, look for ways to eliminate them using perfect square expansions. This avoids having to solve for the individual side lengths $a$ and $c$ separately!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The problem provides the total area, one interior angle ($B$), and the perimeter of a triangle ABC. We need to compute the length of side AC, which is denoted as side $b$ in standard notation.

Step 2: Key Formula or Approach:
We use the standard sine area formula of a triangle to find the product of the adjacent sides:
$$\text{Area} = \frac{1}{2}ac \sin B$$ We then apply the Cosine Rule for angle $B$:
$$\cos B = \frac{a^2 + c^2 - b^2}{2ac}$$ And utilize the algebraic identity $a^2 + c^2 = (a+c)^2 - 2ac$ along with the given perimeter condition $a + b + c = 20$.

Step 3: Detailed Explanation:
1. First, substitute the area and angle $B = 60^\circ$ into the area formula:
$$10\sqrt{3} = \frac{1}{2}ac \sin(60^\circ)$$ Since $\sin(60^\circ) = \frac{\sqrt{3}}{2}$:
$$10\sqrt{3} = \frac{1}{2}ac \left(\frac{\sqrt{3}}{2}\right) \implies 10\sqrt{3} = \frac{\sqrt{3}}{4}ac$$ Cancel out $\sqrt{3}$ from both sides and solve for $ac$:
$$10 = \frac{ac}{4} \implies ac = 40$$ 2. Next, look at the given perimeter equation:
$$a + b + c = 20 \implies a + c = 20 - b$$ 3. Now, set up the Cosine Rule for $\cos(60^\circ)$:
$$\cos(60^\circ) = \frac{a^2 + c^2 - b^2}{2ac}$$ Substitute $\cos(60^\circ) = \frac{1}{2}$ and $ac = 40$:
$$\frac{1}{2} = \frac{(a+c)^2 - 2ac - b^2}{2(40)}$$ $$\frac{1}{2} = \frac{(20-b)^2 - 2(40) - b^2}{80}$$ Cross-multiply to simplify:
$$40 = (400 - 40b + b^2) - 80 - b^2$$ The $b^2$ terms cancel out perfectly:
$$40 = 320 - 40b$$ $$40b = 320 - 40$$ $$40b = 280 \implies b = \frac{280}{40} = 7\ \text{cm}$$ Since side $b$ represents $\ell(\text{AC})$, the length is 7 cm, matching option (D).

Step 4: Final Answer:
The length of side AC is 7 cm, which corresponds to option (D).