Question

A triangle has two sides of lengths 7 and 12. What value could the length of the third side be? Check all that apply.

Answer 1

Answer:

A) $9$

C) $7$

E) $11$

F) $17$

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

$a=7\ units,b=12\ units$

Applying the triangle inequality theorem

1) $7+12 > c$

$19 > c$ -------> $c$ -----> inequality A

2) $7+c > 12$

$c > 12-7$ ------->  $c > 5\ units$ ------> inequality B

The solution for the length of the third side is the interval-----> $(5,19)$

$5 < c < 19$ ------> 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) $9$

Substitute the value in the compound system of inequalities

For $c=9$

$5 < 9 < 19$  -------> is true

therefore

The value of  $9$ can be the length of the third side

case B) $5$

Substitute the value in the compound system of inequalities

For $c=5$

$5 < 5 < 19$  -------> is not true

therefore

The value of  $5$ cannot be the length of the third side

case C) $7$

Substitute the value in the compound system of inequalities

For $c=7$

$5 < 7 < 19$  -------> is true

therefore

The value of  $7$ can be the length of the third side

case D) $3$

Substitute the value in the compound system of inequalities

For $c=3$

$5 < 3 < 19$  -------> is not true

therefore

The value of  $3$ cannot be the length of the third side

case E) $11$

Substitute the value in the compound system of inequalities

For $c=11$

$5 < 11 < 19$  -------> is true

therefore

The value of  $11$ can be the length of the third side

case F) $17$

Substitute the value in the compound system of inequalities

For $c=17$

$5 < 17 < 19$  -------> is true

therefore

The value of  $17$ can be the length of the third side

Answer 2

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