The sides of right-angled triangle containing the right angle are 5x cm and (3x- 1) cm. Calculate the lengths of the hypotenuse of the triangle. If its area is 60 ` cm ^(2)`

Answer

To solve the problem step by step, we will follow the given information and apply the necessary mathematical concepts.

### Step 1: Understand the problem
We have a right-angled triangle with two sides containing the right angle given as \(5x\) cm and \((3x - 1)\) cm. The area of the triangle is given as \(60 \, \text{cm}^2\).

### Step 2: Use the area formula
The area \(A\) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, we can take \(5x\) as the base and \((3x - 1)\) as the height. Therefore, we set up the equation:
\[ 60 = \frac{1}{2} \times 5x \times (3x - 1) \]

### Step 3: Simplify the equation
Multiply both sides by \(2\) to eliminate the fraction:
\[ 120 = 5x \times (3x - 1) \]
Now, distribute \(5x\):
\[ 120 = 15x^2 - 5x \]

### Step 4: Rearrange the equation
Rearranging gives us:
\[ 15x^2 - 5x - 120 = 0 \]

### Step 5: Simplify the equation
Divide the entire equation by \(5\) to simplify:
\[ 3x^2 - x - 24 = 0 \]

### Step 6: Factor the quadratic equation
To factor the quadratic equation, we look for two numbers that multiply to \(-72\) (which is \(3 \times -24\)) and add to \(-1\). The numbers \(-9\) and \(8\) work:
\[ 3x^2 - 9x + 8x - 24 = 0 \]
Group the terms:
\[ 3x(x - 3) + 8(x - 3) = 0 \]
Factoring out \((x - 3)\):
\[ (3x + 8)(x - 3) = 0 \]

### Step 7: Solve for \(x\)
Setting each factor to zero gives:
1. \(3x + 8 = 0 \Rightarrow x = -\frac{8}{3}\) (not valid since \(x\) must be positive)
2. \(x - 3 = 0 \Rightarrow x = 3\)

### Step 8: Calculate the lengths of the sides
Now substitute \(x = 3\) back into the expressions for the sides:
- Side \(AB = 5x = 5 \times 3 = 15 \, \text{cm}\)
- Side \(BC = 3x - 1 = 3 \times 3 - 1 = 9 - 1 = 8 \, \text{cm}\)

### Step 9: Calculate the hypotenuse using Pythagorean theorem
Using the Pythagorean theorem:
\[ h^2 = AB^2 + BC^2 \]
Substituting the values:
\[ h^2 = 15^2 + 8^2 = 225 + 64 = 289 \]
Taking the square root:
\[ h = \sqrt{289} = 17 \, \text{cm} \]

### Final Answer
The length of the hypotenuse of the triangle is \(17 \, \text{cm}\).

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