Question

The altitude to the hypotenuse of a right triangle divides the hypotenuse into segments with lengths in the ratio 1 : 2. The length of the altitude is 8. How long is the hypotenuse?

a. 16

b. 24

c. $4\sqrt{2}$

d. $6\sqrt{6}$

Answer 1

Answer:

a) The length of the hypotenuse = 16 units.

Step-by-step explanation:

Here, let us assume the given right angle triangle is Δ ABC.

Here. AC is the hypotenuse, which is divided in the ratio 1: 2 by the altitude BM.

⇒ AM : MC = 1: 2

Let us assume the common ratio  = x

⇒ AM   =  x and  MC  = 2 x

Also. AM  = 8 units

Solving this by common ratio of sides, we get:

$\implies \frac{2x}{x} = \frac{x}{8} \\\implies 2x^2 = 64\\\implies x^2 = 32\\\implies x = 4\sqrt 2$

Hence, AM  = 4√2  , MC = 2 (4√2)

⇒ AC  = 12√2 units  ≈  16 units

Hence, the length of the hypotenuse = 16 units.