1. **Problem Statement:** We have an isosceles triangle PQR with sides PQ = PR = 5 cm and base QR = 8 cm. A perpendicular from the apex P meets QR at N. We want to find the length of the perpendicular PN. 2. **Formula and Important Rules:** In an isosceles triangle, the altitude from the apex to the base bisects the base. So, N is the midpoint of QR, making QN = NR = $\frac{8}{2} = 4$ cm. 3. **Using the Pythagorean theorem:** Triangle PNQ is a right triangle with hypotenuse PQ = 5 cm, base QN = 4 cm, and height PN (the perpendicular) unknown. 4. **Apply the Pythagorean theorem:** $$PN^2 + QN^2 = PQ^2$$ $$PN^2 + 4^2 = 5^2$$ $$PN^2 + 16 = 25$$ 5. **Solve for PN:** $$PN^2 = 25 - 16 = 9$$ $$PN = \sqrt{9} = 3$$ 6. **Answer:** The length of the perpendicular PN is 3 cm.