The area of a right-angled triangle is 60 cm, and the length of the hypotenuse is 17 cm, find the length of the legs.

Given:

S = 60 cm ^ 2,

c = 17 cm,

a -?

b -?

Decision:

By def. area of a right triangle:

S = a * b / 2 —-> a = 2S / b (1)

By th Pythagoras:

c ^ 2 = a ^ 2 + b ^ 2 —-> b = √ (c ^ 2 – a ^ 2) (2)

(1) —> (2):

a = 2S / √ (c ^ 2 – a ^ 2).

By transforming this expression, we get:

b ^ 4 – c ^ 2b ^ 2 + 4S ^ 2 = 0.

b ^ 2 = x;

x ^ 2 – 289x + 14400 = 0.

D = 83521 – 4 * 14400 = 25921.

x1 = 225;

x2 = 64.

b1 = 15 (cm);

b2 = 8 (cm).

a1 = 2S / b1 = 120/15 = 8 (cm);

a2 = 120/8 = 15 (cm).

Answer: 15 cm and 8 cm or 8 cm and 15 cm.