which value from the list below, when substituted for x, would represent an acute triangle with side lengths…

which value from the list below, when substituted for x, would represent an acute triangle with side lengths x, x + 4, and 20? assume that the longest side of the triangle is of length 20 units. 8 10 12 14

which value from the list below, when substituted for x, would represent an acute triangle with side lengths x, x + 4, and 20? assume that the longest side of the triangle is of length 20 units. 8 10 12 14

Answer

Explanation:

Step1: Recall the condition for an acute triangle

For a triangle with side lengths (a), (b), (c) ((c) is the longest side), the triangle is acute if (a^{2}+b^{2}>c^{2}). Here, (a = x), (b=x + 4), (c = 20). So we need (x^{2}+(x + 4)^{2}>20^{2}). Expand ((x + 4)^{2}=x^{2}+8x + 16). Then the inequality becomes (x^{2}+x^{2}+8x + 16>400), or (2x^{2}+8x-384>0), divide by (2) to get (x^{2}+4x-192>0). Factor the quadratic: (x^{2}+4x-192=(x + 16)(x - 12)>0). The roots are (x=-16) and (x = 12). The solution of the inequality (x^{2}+4x-192>0) for (x>0) (since side - length (x>0)) is (x>12). Also, recall the triangle - inequality: (x+(x + 4)>20) (sum of two shorter sides is greater than the longest side). Simplify (2x+4>20) gives (2x>16) or (x > 8).

Step2: Check each option

  • For (x = 8): (a = 8), (b=8 + 4=12), (c = 20). Then (a^{2}+b^{2}=8^{2}+12^{2}=64 + 144=208), (c^{2}=400). Since (208<400), it is an obtuse triangle.
  • For (x = 10): (a = 10), (b=10 + 4=14), (c = 20). Then (a^{2}+b^{2}=10^{2}+14^{2}=100+196 = 296), (c^{2}=400). Since (296<400), it is an obtuse triangle.
  • For (x = 12): (a = 12), (b=12 + 4=16), (c = 20). Then (a^{2}+b^{2}=12^{2}+16^{2}=144 + 256=400), (c^{2}=400). Since (a^{2}+b^{2}=c^{2}), it is a right - triangle.
  • For (x = 14): (a = 14), (b=14 + 4=18), (c = 20). Then (a^{2}+b^{2}=14^{2}+18^{2}=196+324 = 520), (c^{2}=400). Since (520>400), it is an acute triangle.

Answer:

14