Let the length of the shorter side be x cm. Since the difference between the two legs of the right-angled triangle is 5 cm, the length of the other side will be (x + 5) cm. The hypotenuse is given as 25 cm.

Let the length of the shorter side be x cm. Since the difference between the two legs of the right-angled triangle is 5 cm, the length of the other side will be (x + 5) cm. The hypotenuse is given as 25 cm.

Expanding (x + 5)² gives x² + 10x + 25. Combining like terms yields 2x² + 10x + 25 = 625. Subtracting 625 from both sides gives 2x² + 10x - 600 = 0. Dividing the entire equation by 2 simplifies it to x² + 5x - 300 = 0.

Expanding (x + 5)² gives x² + 10x + 25. Combining like terms yields 2x² + 10x + 25 = 625. Subtracting 625 from both sides gives 2x² + 10x - 600 = 0. Dividing the entire equation by 2 simplifies it to x² + 5x - 300 = 0.

We split the middle term +5x into +20x - 15x because 20 × (-15) = -300. Factoring by grouping gives (x + 20)(x - 15) = 0, which yields x = 15 or x = -20.

Since length cannot be negative, we reject x = -20 and take x = 15. Thus, the two legs are 15 cm and 15 + 5 = 20 cm, and the hypotenuse is 25 cm.

the iPod in our right angle triangle is 25 CM the difference between the length of other two sides is 5 cm find the length of the sides

Answer: The lengths of the two other sides are $15\text{ cm}$ and $20\text{ cm}$ (with hypotenuse $25\text{ cm}$ ).

Step-by-step solution

Step 1: Define the unknown sides

Let the length of the shorter side be $x\text{ cm}$ . Since the difference between the two legs of the right-angled triangle is $5\text{ cm}$ , the length of the other side will be $(x + 5)\text{ cm}$ . The hypotenuse is given as $25\text{ cm}$ .

Step 2: Apply Pythagoras theorem

According to Pythagoras theorem, in a right-angled triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse: $\text{base}^2 + \text{perpendicular}^2 = \text{hypotenuse}^2$ .

Step 3: Form the quadratic equation

Expanding $(x + 5)^2$ gives $x^2 + 10x + 25$ . Combining like terms yields $2x^2 + 10x + 25 = 625$ . Subtracting $625$ from both sides gives $2x^2 + 10x - 600 = 0$ . Dividing the entire equation by $2$ simplifies it to $x^2 + 5x - 300 = 0$ .

Step 4: Solve the quadratic equation by factorisation

We split the middle term $+5x$ into $+20x - 15x$ because $20 \times (-15) = -300$ . Factoring by grouping gives $(x + 20)(x - 15) = 0$ , which yields $x = 15$ or $x = -20$ .

Step 5: Find the side lengths

Since length cannot be negative, we reject $x = -20$ and take $x = 15$ . Thus, the two legs are $15\text{ cm}$ and $15 + 5 = 20\text{ cm}$ , and the hypotenuse is $25\text{ cm}$ .

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