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题目

解题步骤

1. Understand the question

   triangle classification theorems
   signment active
   creating an acute triangle
   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

2. Explanation

   Step1: Recall acute triangle rule

   For an acute triangle with longest side $c$, $a^2 + b^2 > c^2$. Here $c=20$, $a=x$, $b=x+4$.

   Step2: Set up the inequality

   $x^2 + (x+4)^2 > 20^2$
   Expand: $x^2 + x^2 + 8x + 16 > 400$
   Simplify: $2x^2 + 8x - 384 > 0$
   Divide by 2: $x^2 + 4x - 192 > 0$

   Step3: Solve quadratic inequality

   Factor: $(x+16)(x-12) > 0$
   Since side length is positive, $x > 12$.

   Step4: Verify triangle inequality

   Also, $x + (x+4) > 20$ → $2x + 4 > 20$ → $x > 8$. Combine with $x>12$.
   Check options: only 14 satisfies $x>12$.

3. Final answer

   14

答案

Question Analysis

OCR Text

triangle classification theorems
signment active
creating an acute triangle
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

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