Multiple choice

The length of a hypotenuse of a right triangle exceeds the length of its base by $2 cm.$ and exceeds twice the length of the altitude by $1 cm$. Find the length of each side of the triangle (in $cm$).

  1. $6, 8, 10$
  2. $7, 24, 25$
  3. $8, 15, 17$
  4. $7, 40, 41$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let base = b, altitude = a, hypotenuse = h. h = b + 2. h = 2a + 1. So b = 2a - 1. By Pythagoras: (2a-1)^2 + a^2 = (2a+1)^2. 4a^2 - 4a + 1 + a^2 = 4a^2 + 4a + 1. a^2 - 8a = 0. a = 8. Then b = 15, h = 17.

AI explanation

Let the hypotenuse be h, the base be b, and the altitude be a. The problem gives the equations h = b + 2 and h = 2a + 1. Using the Pythagorean theorem, h^2 = a^2 + b^2, and substituting b = h - 2 and a = (h - 1)/2 gives h^2 = ((h - 1)/2)^2 + (h - 2)^2. Solving this quadratic equation yields a hypotenuse of 17 cm. Substituting h = 17 into the initial equations gives the base as 15 cm and the altitude as 8 cm.

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