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.

A.)9
B.)5
C.)7
D.)3
E.)11
F.)17

Answer 1

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

Answer 2

A)
`9`

C)
`7`

E)
`11`

F)
`17`

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<19\ units` -----> 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