The altitude drawn to the hypotenuse of a right triangle divides the hypotenuse into segments such that their lengths are in ratio of 1: 4. if the length of the altitude is 8, find the length of: (a) each segment of the hypotenuse (b) the longer leg of the triangle.

In geometry, it would be always helpful to draw a diagram to illustrate the given problem.

This will also help to identify solutions, or discover missing information.

A figure is drawn for right triangle ABC, right-angled at B.

The altitude is drawn from the right-angled vertex B to the hypotenuse AC, dividing AC into two segments of length x and 4x.

We will be using the first two of the three metric relations of right triangles.

(1) BC^2=CD*CA (similarly, AB^2=AD*AC)

(2) BD^2=CD*DA

(3) CB*BA = BD*AC

From relation (2), we know that

BD^2=CD*DA

substitute values

8^2=x*(4x) => 4x^2=64, x^2=16, x=4

so CD=4, DA=4*4=16 (and AC=16+4=20)

Using relation (1)

AB^2=AD*AC

again, substitute values

AB^2=16*20=320=8^2*5

=>

AB

=sqrt(8^2*5)

=8sqrt(5)

=17.89 (approximately)

step-by-step explanation:

the multiplicative inverse of a fraction a/b is b/a. for the multiplicative inverse of a real number, divide 1 by the number. for example, the reciprocal of 5 is one fifth 1/5 or 0.2, and the reciprocal of 0.25 is 1 divided by 0.25, or 4.