1. **State the problem:** We have a large equilateral triangle formed by 4 identical smaller equilateral triangles. Each small triangle has area $25\sqrt{3}$. We are asked to find the perimeter of the large triangle. 2. **Analyze the figure:** The large triangle is subdivided into 4 small equilateral triangles of equal area and size. This means the large triangle consists of 4 small triangles arranged so that the length of the large triangle's side is twice the side length of a small triangle. 3. **Calculate the side length of the small triangle:** For an equilateral triangle with side length $a$, the area formula is $$ \text{Area} = \frac{\sqrt{3}}{4}a^2. $$ Given the small triangle's area as $25\sqrt{3}$, we set up the equation: $$ 25\sqrt{3} = \frac{\sqrt{3}}{4}a^2. $$ 4. **Solve for $a^2$: ** Divide both sides by $\frac{\sqrt{3}}{4}$: $$ a^2 = 25\sqrt{3} \times \frac{4}{\sqrt{3}} = 25 \times 4 = 100. $$ 5. **Find $a$: ** $$ a = \sqrt{100} = 10. $$ So, each small triangle has side length 10. 6. **Find the side length of the large triangle:** Since the large triangle consists of 4 small triangles arranged so that its side length is twice that of a small triangle, $$ \text{Large side length} = 2 \times 10 = 20. $$ 7. **Calculate the perimeter of the large triangle:** Perimeter is 3 times side length for an equilateral triangle, $$ P = 3 \times 20 = 60. $$ **Final answer:** The perimeter of the large triangle is $60$.