1. **Stating the problem:** We have a large equilateral triangle ABC with sides AB = 7 cm, AC = 5 cm, and BC = x cm. Smaller equilateral triangles are drawn along the sides from corners A and C with sides 3 cm and 2 cm respectively. We need to find the value of $x$. 2. **Important note:** In an equilateral triangle, all sides are equal. Since ABC is equilateral, $AB = BC = AC$. 3. **Using the property of equilateral triangles:** $$AB = BC = AC$$ Given $AB = 7$ cm and $AC = 5$ cm, but this contradicts the property of equilateral triangles because all sides must be equal. 4. **Resolving the contradiction:** Since the problem states ABC is equilateral, the sides must be equal. The given side lengths $AB = 7$ cm and $AC = 5$ cm cannot both be correct for an equilateral triangle. 5. **Assuming the problem means ABC is not equilateral but the smaller triangles are equilateral:** Then the side $BC = x$ is unknown, and we are to find $x$ given the smaller equilateral triangles along the sides. 6. **Given the options for $x$: 3 cm, 5 cm, 7 cm, 8 cm. Since $AB = 7$ cm and $AC = 5$ cm, the side $BC$ must be consistent with the triangle inequality:** - $BC + AC > AB$ implies $x + 5 > 7$ so $x > 2$ - $AB + BC > AC$ implies $7 + x > 5$ always true - $AB + AC > BC$ implies $7 + 5 > x$ so $x < 12$ 7. **From the options, 3, 5, 7, 8 all satisfy the triangle inequality. Since the smaller equilateral triangles have sides 3 cm and 2 cm along the dotted lines, and the large triangle sides are 7 cm and 5 cm, the side $BC$ is likely 7 cm to match $AB$ for symmetry.** **Final answer:** $$x = 7 \text{ cm}$$