Multiple choice

The hypotenuse of a right triangle is $25$ cm.The difference between the lengths of the other two sides of the triangle is $5$ cm. F ind the lengths of these sides.

  1. $20$ cm and $15$ cm
  2. $12$ cm and $13$ cm
  3. $21$ cm and $12$ cm
  4. $13$ cm and $18$ cm
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A Correct answer
Explanation

Let the sides be x and x-5. By the Pythagorean theorem, x^2 + (x-5)^2 = 25^2. Solving x^2 + x^2 - 10x + 25 = 625 gives 2x^2 - 10x - 600 = 0, or x^2 - 5x - 300 = 0. Factoring gives (x-20)(x+15) = 0, so x=20 and the other side is 15.

AI explanation

Let the two unknown sides be x and y. We are given that x^2 + y^2 = 25^2 = 625 and x - y = 5. Using the algebraic identity, (x - y)^2 = x^2 + y^2 - 2xy, we find 5^2 = 625 - 2xy, which gives 2xy = 600. Adding the two equations gives (x + y)^2 = x^2 + y^2 + 2xy = 625 + 600 = 1225, so x + y = 35. Solving the system x + y = 35 and x - y = 5 yields the side lengths of 20 cm and 15 cm.

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