Question

I need to know if it’s a b c d e or f

Answer 1

Let's draw the triangle to better understand the problem:

Let,

x = the length of the third side of the triangle.

Let's recall a property of the sides of a triangle:

"The length of one side of a triangle must always be lesser than the sum of its other two sides"

Let's check each choice to verify which will satisfy the criteria:

a.) x = 5

`\text{ Side 1: 5 }<\text{ 5 + 12 }\rightarrow\text{ 5 }<\text{ 17 ; satisfies the criteria}`
`\text{ Side 2: 5 }<\text{ 5 + 12 }\rightarrow\text{ 5 }<\text{ 17 ; satisfies the criteria}`
`\text{ Side 3: 13 }<\text{ 5 + 5 }\rightarrow\text{ 12 }<\text{ 10 ; does not satisfies the criteria}`

Therefore, x =5 is not applicable.

b.) x = 11

`\text{ Side 1: 11 }<\text{ 5 + 12 }\rightarrow\text{ 11 }<\text{ 17 ; satisfies the criteria}`
`\text{ Side 2: 5 }<\text{ 11 + 12 }\rightarrow\text{ 5 }<\text{ 23 ; satisfies the criteria}`
`\text{ Side 3: 12 }<\text{ 11 + 5 }\rightarrow\text{ 12 }<\text{ 16 ; satisfies the criteria}`

Therefore, x = 11 is applicable.

c.) x = 19

`\text{ Side 2: 19 }<\text{ 5 + 12 }\rightarrow\text{ 19 }<\text{ 17 ; does not satisfies the criteria}`

Therefore, x =19 is not applicable.

d.) x = 9

`\text{ Side 1: 9 }<\text{ 5 + 12 }\rightarrow\text{ 9 }<\text{ 17 ; satisfies the criteria}`
`\text{ Side 2: 5 }<\text{ 9 + 12 }\rightarrow\text{ 5 }<\text{ 21 ; satisfies the criteria}`
`\text{ Side 3: 12 }<\text{ 9 + 5 }\rightarrow\text{ 12 }<\text{ 14; satisfies the criteria}`

Therefore, x = 9 is applicable.

e.) x = 7

`\text{ Side 1: 7 }<\text{ 5 + 12 }\rightarrow\text{ 7 }<\text{ 17 ; satisfies the criteria}`
`\text{ Side 2: 5 }<\text{ 7 + 12 }\rightarrow\text{ 5 }<\text{ 19 ; satisfies the criteria}`
`\text{ Side 3: 12 }<\text{ 7 + 5 }\rightarrow\text{ 12 }<\text{ 12 ; does not satisfies the criteria}`

Therefore, x = 7 is not applicable.