In a right-angled triangle, one leg is 12 mm, its projection onto the hypotenuse is 6 mm.

In a right-angled triangle, one leg is 12 mm, its projection onto the hypotenuse is 6 mm. Find the second leg, the hypotenuse and the projection of the second leg onto the hypotenuse.

The projection of the BC leg onto the AC hypotenuse is the CH segment.
From the right-angled triangle ВСН, according to the Pythagorean theorem, we determine the length of the leg ВН.
BH ^ 2 = BC ^ 2 – CH ^ 2 = 144 – 36 = 108 mm.
Let us prove that right-angled triangles ABN and BCH are similar.
Let the angle BAN = X0, then in the triangle ABN the angle ABN = (90 – X) 0.
In the SVN triangle, the angle SVN = SVA – AVN = (90 – (90 – X) = X0.
Angle SVN = VAN, therefore, right-angled triangles AVN and VSN are similar in acute angle.
Then AH / BC = BC / CH.
AH = BC2 / CH = 108/6 = 18 mm.
The length of the hypotenuse AC = AH + CH = 18 + 6 = 24 cm.
Then AB ^ 2 = AC ^ 2 – BC ^ 2 = 576 – 144 = 432.
AB = 12 * √3 mm.
Answer: The second leg is 12 * √3 mm, the hypotenuse is 24 mm, the projection of the leg is 18 mm.