The ratio of the projections of the legs to the hypotenuse of a right triangle is 0.25.

The ratio of the projections of the legs to the hypotenuse of a right triangle is 0.25. Find the length of the height drawn from the top of the right angle if the length of the hypotenuse is 20cm.

Let’s introduce a variable.

Let m be the length of the projection of one leg of a right-angled triangle, then (4 * m) is the projection of the second leg onto the hypotenuse.

Let’s find the projections of the legs:

m + 4 * m = 20;

m = 4;

4 * m = 16;

We use the Pythagorean theorem for two triangles formed by the height:

a ^ 2 = h ^ 2 + 16;

b ^ 2 = h ^ 2 + 256;

Now we use the Pythagorean theorem for the big triangle:

(5 * m) ^ 2 = a ^ 2 + b ^ 2;

400 = h ^ 2 + 16 + h ^ 2 + 256;

2 * h ^ 2 = 400 – 272;

2 * h ^ 2 = 128;

h ^ 2 = 64;

h = 8 cm.