Question

The perimeter of a right triangle is 24 centimeters. Three times the length of the longer leg minus two times the length of the shorter leg exceeds the hypotenuse by 2 centimeters. What are the lengths
of all three sides?
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Answer 1

Given info : The perimeter of right angled triangle is 24 cm. also 3b - 2b = h + 2 cm , where b is longer leg , a is shorter leg and h is hypotenuse.

To find : the lengths of all sides of triangle are...

solution : perimeter of triangle = a + b + h

⇒24 = a + b + h ....(1)

from question, 3b - 2a = h + 2 ...(2)

from Pythagoras theorem,

h² = a² + b² ...(3)

from equations (1) and (2), we get,

3b - 2a = 24 - a - b + 2

⇒4b - a = 26 ....(4)

from equations (1) and (3) we get,

(24 - a - b)² = a² + b²

and from eq (4) we get,

(24 - 4b + 26 - b)² = (4b - 26)² + b²

⇒50² + 25b² - 500b = 16b² + 676 - 208b + b²

⇒8b² - 292b + 2500 - 676 = 0

⇒8b² - 292b + 1824 = 0

after solving it using quadratic formula you get, b = 8, 57/2

let's take b = 8

then, a = 4b - 26 = 4 × 8 - 26

= 32 - 26 = 6

and h = √(8² + 6²) = 10

Therefore the lengths of sides are 10, 8, 6

Answer 2

$\huge \tt 10,8,6$