102. The sides of right-angled triangle, containing $$ \sqrt{ } 3 $$ the right-angle, are $$ 5 x \mathrm{~cm} $$ and $$ (3 x-1) \mathrm{cm} $$. Calculate the length of the hypotenuse of the triangle, if its area is $$ 60 \mathrm{~cm}^{2} $$.

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To solve the problem, let's break it down step by step.

### Given:
- The triangle is right-angled.
- The two sides containing the right angle are $$5x $$ cm and $$ (3x - 1) $$ cm.
- The area of the triangle is $$60$$ cm².

### Step 1: Use the Area Formula
For a right-angled triangle, the area ($$A$$) is given by:
$$
A = \frac{1}{2} \times \text{base} \times \text{height}
$$
Here, the base and height are the two sides containing the right angle.

$$
60 = \frac{1}{2} \times 5x \times (3x - 1)
$$

### Step 2: Solve for $$x$$
Multiply both sides by 2 to eliminate the fraction:
$$
120 = 5x \times (3x - 1)
$$
$$
120 = 15x^2 - 5x
$$
$$
15x^2 - 5x - 120 = 0
$$

Divide the entire equation by 5 to simplify:
$$
3x^2 - x - 24 = 0
$$

Use the quadratic formula where $$a = 3$$, $$b = -1$$, and $$c = -24$$:
$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$
$$
x = \frac{1 \pm \sqrt{1 + 288}}{6}
$$
$$
x = \frac{1 \pm 17}{6}
$$
Since $$x$$ must be positive:
$$
x = \frac{18}{6} = 3
$$

### Step 3: Determine the Lengths of the Sides
$$
5x = 5 \times 3 = 15 \text{ cm}
$$
$$
3x - 1 = 9 - 1 = 8 \text{ cm}
$$

### Step 4: Find the Hypotenuse
Using the Pythagorean theorem:
$$
\text{Hypotenuse} = \sqrt{15^2 + 8^2} = \sqrt{225 + 64} = \sqrt{289} = 17 \text{ cm}
$$

### **Answer:**
The hypotenuse of the triangle is **17 cm**.
The hypotenuse of the triangle is 17 cm.

The sides of right-angled triangle, containing
the right-angle, are and .
Calculate the length of the hypotenuse of the
triangle, if its area is .

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The sides of right-angled triangle, containing the right-angle, are and . Calculate the length of the hypotenuse of the triangle, if its area is .