If the ratio of the legs of a right triangle is 1 : 2, then the ratio of the corresponding segments of the

hypotenuse made by a perpendicular upon it from the vertex is

Let the altitude h from the right angle divide the hypotenuse into two sections x and y, with x<y.

By similar triangles, we see that

h/x = 2/1
h/y = 1/2

x/y = (h/y)/(h/x) = (1/2)/(2/1) = 1/4

To find the ratio of the corresponding segments of the hypotenuse made by a perpendicular upon it from the vertex, we can use the concept of similar triangles.

Let's say we have a right triangle with legs in the ratio 1 : 2. We can label the lengths of the legs as a and 2a, where 'a' is some positive number.

Now, let's draw a perpendicular from the vertex of the right angle to the hypotenuse. This perpendicular will divide the hypotenuse into two segments.

Let's call the lengths of these segments as x and y. To find the ratio of x to y, we can use the concept of similar triangles.

The two smaller triangles formed by the perpendicular and the legs of the right triangle are similar to the original triangle. This is because they have the same corresponding angles.

The ratio of the corresponding sides of similar triangles is always the same. So, we can set up the following proportion:

(a / x) = ((2a) / y)

We can solve this proportion to find the ratio of x to y:

(a / x) = ((2a) / y)
Cross-multiply:
a * y = (2a) * x
Cancel out the 'a':
y = 2x

Therefore, the ratio of the corresponding segments of the hypotenuse made by a perpendicular upon it from the vertex is 2 : 1.