The leg and hypotenuse of a right triangle are 20 and 52. find the height drawn to the hypotenuse

First of all, you need to make a drawing according to the conditions of the problem:

triangle ABC is rectangular, ∠BAC is straight;
AB = 20;
BC = 52;
AD is the height drawn to the hypotenuse.
Finding the height drawn to the hypotenuse
Consider triangles ABC and ABD. Both of these triangles have one common angle:

∠ABS = ∠ABD.

By definition of cosine, cos ABC = AB / BC.

At the same time, cos ABD = BD / AB.

In this way,

AB / BC = BD / AB;

Multiply the left and right sides of the proportion by AB and swap them:

BD = AB * AB / BC = 20 * 20/52 = 100/13.

Now, knowing the length of the leg BD in the right-angled triangle ABD, you can, using the Pythagorean theorem, find the length of the second leg AD:

AD = √ (AB ^ 2 – BD ^ 2) = √ (20 ^ 2 – (100/13) ^ 2) = √ (400 – 10000/169) = √ (57600/169) = 240/13 ≈ 18, 46.

Answer: The length of the height of a right-angled triangle, drawn to the hypotenuse, is 18.46.