The height of a right-angled triangle with an acute angle a, drawn to the hypotenuse, equals h. Find the hypotenuse of the triangle.

Since CH is the height of the triangle ABC, the triangles AСН and BCH are rectangular.

Then in a right-angled triangle ACC, tgα = CH / AН.

AH = h / tgα.

In the ACN triangle, the ACН angle = (90 – α) 0.

Then in the right-angled triangle СВН the angle BCH = (90 – (90 – α)) = α0.

In a right-angled triangle BCH, tgα = BH / CH.

BH = h * tgα.

Then AB = h / tgα + h * tgα = h * (ctgα + tgα) = h * (Cosα / Sinα + Sinα / Cosα) = h * ((Cos2α + Sin2α) / Cosα * Sinα) = h / (Cosα * Sinα).

Answer: The length of the hypotenuse is = h / (Cosα * Sinα).