1. **Problem Statement:** We are given a triangle with sides 7 cm, 13 cm, and 12 cm. We need to find the length of the perpendicular from the vertex opposite the side of length 12 cm to that side. 2. **Formula and Important Rules:** The perpendicular height $h$ to a side of length $b$ in a triangle can be found using the area $A$ of the triangle: $$h = \frac{2A}{b}$$ We first find the area using Heron's formula: $$s = \frac{a+b+c}{2}$$ $$A = \sqrt{s(s-a)(s-b)(s-c)}$$ where $a$, $b$, and $c$ are the side lengths. 3. **Calculate the semi-perimeter $s$:** $$s = \frac{7 + 13 + 12}{2} = \frac{32}{2} = 16$$ 4. **Calculate the area $A$ using Heron's formula:** $$A = \sqrt{16(16-7)(16-13)(16-12)} = \sqrt{16 \times 9 \times 3 \times 4}$$ $$= \sqrt{1728} = 12\sqrt{12} = 12 \times 2\sqrt{3} = 24\sqrt{3}$$ 5. **Calculate the perpendicular height $h$ to the side of length 12 cm:** $$h = \frac{2A}{12} = \frac{2 \times 24\sqrt{3}}{12} = \frac{48\sqrt{3}}{12} = 4\sqrt{3}$$ 6. **Final answer:** The length of the perpendicular is $4\sqrt{3}$ cm, approximately 6.93 cm.