In a right-angled triangle ABC, the length of the BC leg is 13 cm, and the length of the height CD

In a right-angled triangle ABC, the length of the BC leg is 13 cm, and the length of the height CD held with the hypotenuse AB is 12 cm. Calculate the length of the projection of the BC leg onto the hypotenuse and the length of the AC leg.

Given: ∆ABC – rectangular (<C = 90 °), BC = 13 cm, height CD = 12 cm (drawn to AB).
Find: BD, AC.
∆CDB is rectangular because CD – perpendicular. Behind the Pythagorean theorem:
BC² = CD² + BD².
Hence:
BD² = BC² – CD²,
BD² = 13² – 12² = 169 – 144 = 25 (cm),
BD = √25 = 5 (cm).
For the property of the height drawn to the hypotenuse:
CD = √AD * √BD,
CD² = AD * BD.
Hence:
AD = CD² / BD,
AD = 12² / 5 = 144/5 = 28.8 (cm).
Find the hypotenuse AB:
AB = AD + BD,
AB = 28.8 + 5 = 33.8 (cm).
Behind the Pythagorean theorem:
AB² = AC² + BC².
Hence:
AC² = AB² – BC²,
AC² = 33.8² – 13² = 1142.44 – 169 = 973.44 (cm),
AC = √973.44 = 31.2 (cm).
Answer: AC = 31.2 cm; BD = 5 cm.