We have ABC,
And it’s 3 sides are :
AB
BC
AC
Consider AB is the shortest and AC is the longest side so we have :
AB = 3x
BC = ? (Y)
AC = 5x
Because it is a right triangle, we have : AC^2 = BC^2 + AB^2
(5x)^2 = (3x)^2 + y^2 y^2 = 25x^2 - 9x^2 —> y = 4x

Area : 1/2 * AB * BC = 1/2 * 3x * 4x = 6x^2
We have area between 50 and 150 so:
50 < 6x^2 < 150
25 < 3x^2 < 75
8.3 < x^2 < 25
2.88 < x < 5

So the smallest side which is 3x, is between 8.64 and 15.
C, D and E can be answers to this question.

I made the shortest side a, the longest side b and the remaining side x . We want to find what a would be if the area is between 50 and 150. We know that side b is the hypotenuse as it is the longest side on a right triangle.

a/b = 3/5 ----> 5a = 3b ---> b= 5a/3

We know that the area is the two legs (a and x) divided by 2: xa/2

a^2 + x^2 = b^2 get x in terms of a

a^2 + x^2 = ( 5a/3)^2

a^2 + x^2 = 25a^2/9

x^2 = (25a^2/9) - a^2

x^2 = (25a^2/9) - 9a^2/9

x^2 = 16a^2/9 take the square root ---> x = 4a/3

plug into area formula

(4a/3)(a) (1/2) ----> 4a^2/6 = area

Now we need to solve for a (shortest side) for the upper and lower limits.

4a^2/6 > 50
4a^2 > 300
a^2 > 75
a > sqr 75 (~ 8.7)


4a^2/6 < 150
4a^2 < 900
a^2 < 225
a < 15

From this, we can find that 8.7 < a< 15

The only answers that fall in that range are C (9) and D(12)

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