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triangle classification theorems signment active creating an acute tria…

14

Categoría: geometry Actualizado: 2026-02-09

Pregunta

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

Pasos de solución

  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

Respuesta

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

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Question 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

Show OCR extraction 
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

Temas relacionados

mathematicsgeometrymultiple choice, calculationhigh schoolturns-1

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