Question

Triangle Inequality Theorem Quick Check
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Question
Determine the length of the third side of a triangle from the choices provided if the first two sides measure 10 and 12.(1 point)
Responses

15
15

22
22

24
24

30
30
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Answer 1

To determine the length of the third side of a triangle when the first two sides are given (10 and 12), we can use the Triangle Inequality Theorem. This 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.

Let’s denote the sides as follows:
- Side a = 10
- Side b = 12
- Side c = the third side (unknown)

According to the Triangle Inequality Theorem, we need to satisfy the following conditions:

1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

Now, substituting \( a \) and \( b \) into the inequalities:

1. \( 10 + 12 > c \) → \( 22 > c \)
2. \( 10 + c > 12 \) → \( c > 2 \)
3. \( 12 + c > 10 \) → \( c > -2 \) (this condition is always satisfied since \( c > 2 \))

Combining the useful inequalities from above, we find:
\( 2 < c < 22 \)

Now, looking at the provided choices:
- 15 (Valid)
- 22 (Invalid, as it must be less than 22)
- 24 (Invalid, as it exceeds 22)
- 30 (Invalid, as it exceeds 22)

The only valid choice we have is **15**.

Thus, the length of the third side is **15**.