Questions: Дві сторони трикутника 6 см і 4 см, кут між ними - 120 градусів. Знайдіть довжину третьої сторони трикутника. √76 см √(52-24√3) см 2√7 см √(52+24√3) см

Transcript text: Дві сторони трикутника $6 с м$ і $4 с м$, кут між ними - $120^{\circ}$. Знайдіть довжину третьоі сторони трикутника. $\sqrt{76} \mathrm{~cm}$ $\sqrt{52-24 \sqrt{3}} \mathrm{~cm}$ $2 \sqrt{7} \mathrm{~cm}$ $\sqrt{52+24 \sqrt{3}} \mathrm{~cm}$

Solution

Solution Steps

To find the length of the third side of a triangle when two sides and the included angle are given, we can use the Law of Cosines. The Law of Cosines states that for a triangle with sides $a$, $b$, and $c$, and an angle $\gamma$ opposite side $c$, the formula is:

\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(\gamma) \]

In this problem, the sides are $6 \, \text{cm}$ and $4 \, \text{cm}$, and the angle between them is $120^\circ$. We will use this formula to calculate the length of the third side.

Step 1: Given Values

We are given two sides of a triangle: $ a = 6 \, \text{cm} $ and $ b = 4 \, \text{cm} $, with the included angle $ \gamma = 120^\circ $.

Step 2: Convert Angle to Radians

To use the Law of Cosines, we first convert the angle from degrees to radians: \[ \gamma = 120^\circ = \frac{2\pi}{3} \, \text{radians} \approx 2.0944 \, \text{radians} \]

Step 3: Apply the Law of Cosines

Using the Law of Cosines, we calculate the square of the length of the third side $ c $: \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(\gamma) \] Substituting the values: \[ c^2 = 6^2 + 4^2 - 2 \cdot 6 \cdot 4 \cdot \cos\left(\frac{2\pi}{3}\right) \] Calculating each term: \[ c^2 = 36 + 16 - 48 \cdot \left(-\frac{1}{2}\right) \] \[ c^2 = 36 + 16 + 24 = 76 \]

Step 4: Calculate the Length of the Third Side

Now, we take the square root to find $ c $: \[ c = \sqrt{76} \approx 8.7178 \, \text{cm} \]