In an ABC triangle, angle C is 90 degrees. The legs of the triangle are 16 and 12.

In an ABC triangle, angle C is 90 degrees. The legs of the triangle are 16 and 12. Find the length AK of the projection of the AC leg onto the hypotenuse.

Let us determine, using the Pythagorean theorem, the length of the hypotenuse AB.
AB2 = AC^2 + CB^2 = 16^2 + 12^2 = 256 + 144 = 400.
AB = 20 cm.
Let the length of the segment BH = X cm, then the length of the segment AH = (20 – X) cm.
In two right-angled triangles, ASN and BCH, we express the height of the CH.
CH2 = AC^2 – AH^2 = 16^2 – (20 – X) ^2 = 256 – 400 + 40 * X – X2 = -144 + 40 * X – X2.
CH2 = BC^2 – BH^2 = 144 – X2.
Then: -144 + 40 * X – X^2 = 144 – X2.
40 * X = 288.
X = BH = 288/40 = 7.2 cm.
Then AH = 20 – 7.2 = 12.8 cm.
Answer: The projection of the AC leg is 12.8 cm.