Multiple choice

A right triangle has a perimeter of $48 cm$ and an area of $96 \displaystyle cm^{2}$. Find its hypotenuse (in cm).

  1. $18$
  2. $20$
  3. $24$
  4. $27$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Perimeter = a+b+c = 48. Area = 0.5ab = 96, so ab = 192. c^2 = a^2+b^2 = (a+b)^2 - 2ab = (48-c)^2 - 2(192) = 2304 - 96c + c^2 - 384. 96c = 1920, c = 20.

AI explanation

Using the formulas for area and perimeter gives the equations 96 = 0.5 * a * b and 48 = a + b + c. Squaring the perimeter equation gives 2304 = a^2 + b^2 + c^2 + 2ab + 2c(a + b), and substituting the known values of a^2 + b^2 = c^2 and 2ab = 384 simplifies this to c^2 + 48c = 1920. Solving the quadratic equation c^2 + 48c - 1920 = 0 gives the hypotenuse c = 20 cm.

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