Question

55 POINTS!!! The hypotenuse of a right triangle is 5cm more than 4 times the length of the shortest leg. The longer leg and the hypotenuse are consecutive integers. Find all lengths. ​

Answer 1

Answer:

Short leg is 9.

Hypotenuse is  41.

Long leg is 40.

Step-by-step explanation:

Shortest leg  is x.

Hypotenuse is (4x + 5).

Longer leg is shorter than hypotenuse, and they are consecutive numbers,

so longer leg is (4x + 5 - 1) = (4x + 4).

Using Pythagorean theorem

x² + (4x + 4)² = (4x + 5)²

x² + 16x² + 2*4x*4 + 4² = 16x² + 2*4x*5 + 5²

x² + 32x + 16 = 40x + 25

(x - 4)² = 5²

x - 4 = 5 or  x - 4 = - 5

x = 9 , or x = - 1

Leg cannot be negative.

So, short leg is 9.

Hypotenuse (4x+5) = (4*9 + 5) = 41

Long leg is 41 - 1 = 40

Check:

9² + 40² = 41²

1681 = 1681 True