The length of hypotenuse of a right angled triangle is `3sqrt(10)`. If among two perpendicular lines, smallest one is tripled and bigger one is double bled, the hypotenuse of new right angled triangle thus formed is `9sqrt(5)` unit. The length of smallest and the bigger sides are respectively.

Options

A. a)5 unit, 9 unit
B. b)5 unit, 6 unit
C. c)3 unit, 9 unit
D. d)3 unit, 6 unit

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the given information
We know the hypotenuse of the original right-angled triangle is \(3\sqrt{10}\). We also know that the lengths of the two perpendicular sides (let's call them \(a\) and \(b\)) can be found using the Pythagorean theorem:

\[ c^2 = a^2 + b^2 \]

where \(c\) is the hypotenuse.

### Step 2: Calculate the square of the hypotenuse
We calculate the square of the hypotenuse:

\[ c^2 = (3\sqrt{10})^2 = 9 \times 10 = 90 \]

So, we have:

\[ a^2 + b^2 = 90 \quad \text{(1)} \]

### Step 3: Analyze the new triangle
In the new triangle, the smallest side \(a\) is tripled and the larger side \(b\) is doubled. The new hypotenuse is given as \(9\sqrt{5}\).

Using the Pythagorean theorem again for the new triangle, we have:

\[ (3a)^2 + (2b)^2 = (9\sqrt{5})^2 \]

Calculating the square of the new hypotenuse:

\[ (9\sqrt{5})^2 = 81 \times 5 = 405 \]

So, we have:

\[ (3a)^2 + (2b)^2 = 405 \quad \text{(2)} \]

### Step 4: Expand the equation (2)
Expanding the equation (2):

\[ 9a^2 + 4b^2 = 405 \quad \text{(3)} \]

### Step 5: Solve the system of equations
Now we have two equations:

1. \(a^2 + b^2 = 90\) (from equation (1))
2. \(9a^2 + 4b^2 = 405\) (from equation (3))

From equation (1), we can express \(b^2\) in terms of \(a^2\):

\[ b^2 = 90 - a^2 \]

Substituting \(b^2\) into equation (3):

\[ 9a^2 + 4(90 - a^2) = 405 \]

Expanding this:

\[ 9a^2 + 360 - 4a^2 = 405 \]

Combining like terms:

\[ 5a^2 + 360 = 405 \]

Subtracting 360 from both sides:

\[ 5a^2 = 45 \]

Dividing by 5:

\[ a^2 = 9 \]

Taking the square root:

\[ a = 3 \]

### Step 6: Find \(b\)
Now substituting \(a = 3\) back into equation (1) to find \(b\):

\[ 3^2 + b^2 = 90 \]

\[ 9 + b^2 = 90 \]

Subtracting 9 from both sides:

\[ b^2 = 81 \]

Taking the square root:

\[ b = 9 \]

### Final Answer
The lengths of the smallest and the bigger sides are respectively \(3\) units and \(9\) units.

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