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In an isosceles triangle ABC, with AB = AC, the altitude from A meets BC at D. If AB = 6 cm and AD = 4 cm, find the length of BD.

In an equilateral triangle ABC, the altitude from A meets BC at D. If AD = 4 cm, find the length of BC.

In a triangle ABC, the median from A meets BC at D. If AB = 8 cm, AC = 6 cm, and BD = 3 cm, find the length of AD.

In a triangle PQR, the median from P meets QR at S. If PQ = 10 cm, PR = 12 cm, and PS = 6 cm, find the length of QS.

In a right-angled triangle XYZ, the altitude from X meets YZ at P. If XY = 6 cm, XZ = 8 cm, and XP = 4.8 cm, find the length of YP.