triangle classification theorems signment active creating an acute tria…
14
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
Solution Steps
-
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 -
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 $x>12$. -
Final answer
14
Answer
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 $x>12$.
Answer
14
Question Image
Click to enlargeQuestion Analysis
Subject
mathematics
Sub Subject
geometry
Education Level
high school
Difficulty
unspecified
Question Type
multiple choice, calculation
Multi Question
No
Question Count
1
Analysis Status
completed
Analyzed At
2026-02-09T20:20:51
OCR Text
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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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