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A triangle has two sides of lengths 7 and 12. What value could the length of the third side be? Check all that apply.
A.)9
B.)5
C.)7
D.)3
E.)11
F.)17

Respuesta :

wolf1728
Third side must be greater than (12 -7) or 5 and less than (12 + 7) or 19
So, third side could be A C E or F

calculista

Answer:

A) [tex]9[/tex]

C) [tex]7[/tex]

E) [tex]11[/tex]

F) [tex]17[/tex]

Step-by-step explanation:

we know that

The Triangle Inequality Theorem, states  that the sum of the lengths of any two sides of a triangle is greater than the length of the third side

Let

a,b,c ------>  the length sides of the triangle

in this problem we have

[tex]a=7\ units,b=12\ units[/tex]

Applying the triangle inequality theorem

1) [tex]7+12 > c[/tex]

[tex]19 > c[/tex] -------> [tex]c<19\ units[/tex] -----> inequality A

2) [tex]7+c > 12[/tex]

[tex]c > 12-7[/tex] ------->  [tex]c > 5\ units[/tex] ------> inequality B

The solution for the length of the third side is the interval-----> [tex](5,19)[/tex]

[tex]5 < c < 19[/tex] ------> compound system of inequalities

we're going to verify all the cases

If a value could be the length of the third side, then the value must be satisfy  the compound system of inequalities

case A) [tex]9[/tex]

Substitute the value in the compound system of inequalities

For [tex]c=9[/tex]

[tex]5 < 9 < 19[/tex]  -------> is true

therefore

The value of  [tex]9[/tex] can be the length of the third side

case B) [tex]5[/tex]

Substitute the value in the compound system of inequalities

For [tex]c=5[/tex]

[tex]5 < 5 < 19[/tex]  -------> is not true

therefore

The value of  [tex]5[/tex] cannot be the length of the third side

case C) [tex]7[/tex]

Substitute the value in the compound system of inequalities

For [tex]c=7[/tex]

[tex]5 < 7 < 19[/tex]  -------> is true

therefore

The value of  [tex]7[/tex] can be the length of the third side

case D) [tex]3[/tex]

Substitute the value in the compound system of inequalities

For [tex]c=3[/tex]

[tex]5 < 3 < 19[/tex]  -------> is not true

therefore

The value of  [tex]3[/tex] cannot be the length of the third side

case E) [tex]11[/tex]

Substitute the value in the compound system of inequalities

For [tex]c=11[/tex]

[tex]5 < 11 < 19[/tex]  -------> is true

therefore

The value of  [tex]11[/tex] can be the length of the third side

case F) [tex]17[/tex]

Substitute the value in the compound system of inequalities

For [tex]c=17[/tex]

[tex]5 < 17 < 19[/tex]  -------> is true

therefore

The value of  [tex]17[/tex] can be the length of the third side