In a right-angled triangle, the leg ratio is 1: 2. Find the ratio of the projection of these legs to the hypotenuse.

Let’s build the height BH, then the segments AH and CH are the projections of the legs AB and BC onto the hypotenuse.

Let us prove the similarity of triangles ABН and BCH.

Let the angle BCH = X0, then the angle BCH = (90 – X) 0.

In triangle ABН, angle ABN = (90 – СBН) = (90 – (90 – X) = X0.

Then triangles ABH and СBH are similar in acute angle.

Then AB / AH = BC / BH.

By condition, AB = 2 * BC, then: 2 * BC / AH = BC / BN.

AH = 2 * BH.

BH = AH / 2. (1).

BC / CH = AB / BH.

ВС / СН = 2 * ВС / ВН.

BH = 2 * CH. (2).

Equate Equations 1 and 2.

AH / 2 = 2 * CH.

CH / AH = 1/4.

Answer: The ratio of the projections of the legs is 1/4.