A In a right-angled triangle, one of the perpendicular sides is 2 cm longer than
the other. The hypotenuse is more than twice the length of the shortest side.
Determine the possible lengths for the shortest side.

The possible lengths of the shortest side, 'y', are 0 cm < y < (1 + √3) cm

Step-by-step explanation:

The given parameters of the right triangle are;

The length of one of the perpendicular sides = 2 cm longer than the other

The length of the hypotenuse side > 2 × The length of the shortest side

Let 'x' represent the length of one of the perpendicular sides, let 'y', represent the length of the shortest perpendicular side, and let 'h', represent the hypotenuse side, we have;

x = 2 + y...(1)

h > 2·y...(2)

From equation (1), we get;

y = x - 2

By Pythagoras theorem, the hypotenuse side, 'h', is given as follows;

h = √(x² + y²) > 2 × y

Simplifying the above equation by plugging in x = y + 2, gives

h = √((y + 2)² + y²) > 2 × y

h² = 2·y² + 4·y + 4 > 4·y²

∴ 2·y² - 4·y - 4  <  0

2·(y - 1 + √3)·(y - (1 + √3)) < 0

y < 1 - √3 < 0, or y < 1 + √3

Therefore, the possible lengths of the shortest side of the right triangle, y are;

0 cm < y < (1 + √3) cm.

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