Let the lengths of the two legs of the right triangle be a and b, and let the hypotenuse be c = 25 cm. Since the perimeter is the sum of all 3 sides, we have a + b + 25 = 60, which simplifies to a + b = 35. If we let one side be x cm, the other side is (35 - x) cm.

Let the lengths of the two legs of the right triangle be a and b, and let the hypotenuse be c = 25 cm. Since the perimeter is the sum of all 3 sides, we have a + b + 25 = 60, which simplifies to a + b = 35. If we let one side be x cm, the other side is (35 - x) cm.

Expanding (35 - x)² gives 1225 - 70x + x². Combining like terms yields 2x² - 70x + 1225 = 625. Subtracting 625 from both sides gives 2x² - 70x + 600 = 0, and dividing the entire equation by 2 simplifies it to x² - 35x + 300 = 0.

Expanding (35 - x)² gives 1225 - 70x + x². Combining like terms yields 2x² - 70x + 1225 = 625. Subtracting 625 from both sides gives 2x² - 70x + 600 = 0, and dividing the entire equation by 2 simplifies it to x² - 35x + 300 = 0.

When one side is 20 cm, the other side is 35 - 20 = 15 cm. Conversely, if one side is 15 cm, the other is 20 cm. Hence, the lengths of the other two sides are 15 cm and 20 cm.

19. The perimeter of a right triangle is $60\text{ cm}$ and its hypotenuse is $25\text{ cm}$ . Find the lengths of other two sides of the triangle.

Answer: The lengths of the other two sides of the triangle are $15\text{ cm}$ and $20\text{ cm}$ .

Step-by-step solution

Step 1: Express the sides in terms of a single variable

Let the lengths of the two legs of the right triangle be $a$ and $b$ , and let the hypotenuse be $c = 25\text{ cm}$ . Since the perimeter is the sum of all 3 sides, we have $a + b + 25 = 60$ , which simplifies to $a + b = 35$ . If we let one side be $x\text{ cm}$ , the other side is $(35 - x)\text{ cm}$ .

Step 2: Apply Pythagoras theorem to form a quadratic equation

By Pythagoras theorem, the sum of the squares of the legs of a right triangle is equal to the square of its hypotenuse. Substituting the lengths $x$ , $35 - x$ , and $25$ , we obtain the equation $x^2 + (35 - x)^2 = 25^2$ .

Step 3: Expand and simplify into standard quadratic form

Expanding $(35 - x)^2$ gives $1225 - 70x + x^2$ . Combining like terms yields $2x^2 - 70x + 1225 = 625$ . Subtracting $625$ from both sides gives $2x^2 - 70x + 600 = 0$ , and dividing the entire equation by $2$ simplifies it to $x^2 - 35x + 300 = 0$ .

Step 4: Solve the quadratic equation by factorisation

We find two numbers whose sum is $-35$ and whose product is $300$ . These numbers are $-20$ and $-15$ . Factoring by grouping gives $(x - 20)(x - 15) = 0$ , which yields the solutions $x = 20$ or $x = 15$ .

Step 5: State the lengths of the two sides

When one side is $20\text{ cm}$ , the other side is $35 - 20 = 15\text{ cm}$ . Conversely, if one side is $15\text{ cm}$ , the other is $20\text{ cm}$ . Hence, the lengths of the other two sides are $15\text{ cm}$ and $20\text{ cm}$ .

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19. The perimeter of a right triangle is 60 cm60\text{ cm}60 cm and its hypotenuse is 25 cm25\text{ cm}25 cm. Find the lengths of other two sides of the triangle.