In right triangle, ABC, the ratio of the longest side to the shortest

Let longest side=5x and shortest side=3x
Third side= [(5x)^2 - (3x)^2]^1/2= 4x {Pythagoras Theorm}

Area= 1/2*4x*3x=6x^2

50<6x^2<150
75<9x^2<225
8.6<3x<15

Shortest side could be 9 or 12

ABC is right angled triangle. Now ratio of sides- 5x:4x:3x

50 < (1/2)*(3x)*(4X) < 150
100< 12 (x^2)< 300
8.abc < (x^2) < 25

x can be 3,4

I 9 possible
II 12 possible
III 15 not possible. Answer D.

If the triangle is a right triangle then the ratio of the sides should necessarily be 3:4:5 as it’s already said that the ratio of shortest side to longest side is 3:5

Now,
If the two smaller sides are 3x and 4x,
Area = 0.5*3x*4x = 6x^2

Now, let’s evaluate the options:
If the smaller side is 9, the second side has to be 12 as the ratio is 3:4
So, the area is 0.5*9*12 = 54 which is within the given range.
So, 9 can be the shortest side.

If the smaller side is 12, the second side has to be 16 as the ratio is 3:4
So, the area is 0.5*12*16 = 96 which is within the given range.
So, 12 can be the shortest side.

If the smaller side is 15, the second side has to be 20 as the ratio is 3:4
So, the area is 0.5*15*20 = 150 which is not within the given range.
So, 15 cannot be the shortest side.

Hence, only 9 and 12 are the possible values.

Marking D safely and confidently.
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Kudos will help me to reach there

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so let the ratio be 5x and 3x

since it is a right-angled triangle, so we have the third side as 4x ( \((3x^2 + 4x^2 = 5x^2\)) )

area becomes = \(1/2 * 3x * 4x\) = 6x^2


given
50 < 6x^2 < 150

we would need to key in integer values
x could be 2 and 3 only to satisfy the condition.

so answer is D

Since triangle ABC is a right triangle with ratio of the longest side to the shortest side of 5 to 3, it must be a 3-4-5 right triangle. Let’s analyze the Roman numerals now (keep in mind that the area of a right triangle is ½ of the product of the length of the two legs).

I. 9

If the shortest side (or leg) is 3 x 3 = 9, then the other leg is 4 x 3 = 12. Therefore, the area of the triangle would be ½(9)(12) = 54. This works since 54 is between 50 and 150.

II. 12

If the shortest side (or leg) is 3 x 4 = 12, then the other leg is 4 x 4 = 16. Therefore, the area of the triangle would be ½(12)(16) = 96. This works since 96 is between 50 and 150.

III. 15

If the shortest side (or leg) is 3 x 5 = 15, then the other leg is 4 x 5 = 20. Therefore, the area of the triangle would be ½(15)(20) = 150. This doesn’t work since 150 is NOT between 50 and 150.

Answer: D
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