Answer:
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.
