Find the hypotenuse of a right-angled triangle if 1 of the legs and its projection onto the hypotenuse is 4 cm and 2 cm.

The projection of the BC leg onto the AC hypotenuse is the CH segment, the length of which is 2 cm.

The first way.

The BCH triangle is rectangular, then, according to the Pythagorean theorem, we determine the length of the BH leg.

BH ^ 2 = BC ^ 2 – CH ^ 2 = 16 – 4 = 12.

BH = 2 * √3 cm.

In a right-angled triangle ABC, the BH segment is the height drawn from the vertex of the right angle to the hypotenuse, then BH2 = AH * CH.

12 = AH * 2.

AH = 12/2 = 6 cm, then AC = AH + CH = 6 + 2 = 8 cm.

Second way.

In a right-angled triangle ВСН, the length of the СН leg is half the hypotenuse BC, then the angle СНН = 30. The angle ВСН = 90 – 30 = 60.

Then the angle BAC = 90 – 60 = 30.

The BC leg lies against an angle of 30, then AC = 2 * BC = 2 * 4 = 8 cm.

Answer: The length of the hypotenuse is 8 cm.