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Can the sides of a triangle have lengths $1$ , $9$ , and $9$ ? $\newline$ Choices: $\newline$ (A) yes $\newline$ (B) no

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Triangle inequality

Full solution

Q. Can the sides of a triangle have lengths

1

,

9

, and

9

?

\newline

Choices:

\newline

(A) yes

\newline

(B) no

Check Triangle Inequality Theorem: To determine if these can be the sides of a triangle, we need to check the triangle inequality theorem. The theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Check $1 + 9 > 9$ : Check if $1 + 9 > 9$ . Calculation: $1 + 9 = 10$ , and $10 > 9$ , so this part is true.
Check $1 + 9 > 9$ : Check if $1 + 9 > 9$ . Calculation: $1 + 9 = 10$ , and $10 > 9$ , so this part is also true.
Check $9 + 9 > 1$ : Check if $9 + 9 > 1$ . Calculation: $9 + 9 = 18$ , and $18 > 1$ , so this part is true too.
Confirm Triangle Formation: Since all conditions of the triangle inequality theorem are met, the sides $1$ , $9$ , and $9$ can indeed form a triangle.

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