Properties of Triangle Formed by Three Vectors

To determine which statements are true, we use the properties of vector addition in a triangle and vector products.

Since $\vec{a} = \vec{QR}$, $\vec{b} = \vec{RP}$, and $\vec{c} = \vec{PQ}$ form the sides of $\Delta PQR$, their sum is zero:

$$\vec{a} + \vec{b} + \vec{c} = \vec{0} \implies \vec{a} = -(\vec{b} + \vec{c})$$

We are given:

$$|\vec{a}| = 12, \quad |\vec{b}| = 4\sqrt{3}, \quad \text{and} \quad \vec{b} \cdot \vec{c} = 24$$

Step 1: Finding $|\vec{c}|^2$

Taking the magnitude squared of $\vec{a} = -(\vec{b} + \vec{c})$:

$$|\vec{a}|^2 = |\vec{b} + \vec{c}|^2 = |\vec{b}|^2 + |\vec{c}|^2 + 2(\vec{b} \cdot \vec{c})$$

Substituting the given values:

$$12^2 = (4\sqrt{3})^2 + |\vec{c}|^2 + 2(24)$$ $$144 = 48 + |\vec{c}|^2 + 48$$ $$144 = 96 + |\vec{c}|^2 \implies |\vec{c}|^2 = 48$$

Step 2: Evaluating Statements (A) and (B)

For statement (A):

$$\frac{|\vec{c}|^2}{2} - |\vec{a}| = \frac{48}{2} - 12 = 24 - 12 = 12$$

Thus, statement (A) is true.

For statement (B):

$$\frac{|\vec{c}|^2}{2} + |\vec{a}| = \frac{48}{2} + 12 = 24 + 12 = 36 \neq 30$$

Thus, statement (B) is false.

Step 3: Evaluating Statement (D)

Using $\vec{c} = -(\vec{a} + \vec{b})$:

$$|\vec{c}|^2 = |\vec{a} + \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + 2(\vec{a} \cdot \vec{b})$$

Substituting the values:

$$48 = 12^2 + (4\sqrt{3})^2 + 2(\vec{a} \cdot \vec{b})$$ $$48 = 144 + 48 + 2(\vec{a} \cdot \vec{b})$$ $$2(\vec{a} \cdot \vec{b}) = -144 \implies \vec{a} \cdot \vec{b} = -72$$

Thus, statement (D) is true.

Step 4: Evaluating Statement (C)

Since $\vec{a} + \vec{b} + \vec{c} = \vec{0}$, taking the cross product with $\vec{a}$:

$$\vec{a} \times (\vec{a} + \vec{b} + \vec{c}) = \vec{0} \implies \vec{a} \times \vec{b} + \vec{a} \times \vec{c} = \vec{0} \implies \vec{a} \times \vec{b} = \vec{c} \times \vec{a}$$

Therefore:

$$|\vec{a} \times \vec{b} + \vec{c} \times \vec{a}| = |2(\vec{a} \times \vec{b})| = 2|\vec{a} \times \vec{b}|$$

Using Lagrange's identity:

$$|\vec{a} \times \vec{b}|^2 = |\vec{a}|^2 |\vec{b}|^2 - (\vec{a} \cdot \vec{b})^2$$ $$|\vec{a} \times \vec{b}|^2 = (144)(48) - (-72)^2 = 6912 - 5184 = 1728$$ $$|\vec{a} \times \vec{b}| = \sqrt{1728} = 24\sqrt{3}$$

Hence:

$$|\vec{a} \times \vec{b} + \vec{c} \times \vec{a}| = 2 \times 24\sqrt{3} = 48\sqrt{3}$$

Thus, statement (C) is true.

Conclusion

The correct statements are A, C, and D.