Question

A triangle has two sides that each measure 8 centimeters and 12 centimeters.



Choose ,begin emphasis,all,end emphasis, the possible measures of the third side of the triangle.

Answer options with 5 options
A.
3 centimeters

B.
4 centimeters

C.
5 centimeters

D.
8 centimeters

E.
20 centimeters

Answer 1

To determine the possible measures of the third side of a triangle given two sides measuring 8 cm and 12 cm, 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 the lengths of the two known sides be \( a = 8 \) cm and \( b = 12 \) cm, and let \( c \) be the length of the third side. According to the triangle inequality, the following conditions must be satisfied:

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

Substituting the values of \( a \) and \( b \):

1. \( 8 + 12 > c \) → \( 20 > c \) → \( c < 20 \)
2. \( 8 + c > 12 \) → \( c > 4 \)
3. \( 12 + c > 8 \) → \( c > -4 \) (This condition does not affect the range since any positive length will satisfy it)

From the first two inequalities, we must have:

\[ 4 < c < 20 \]

Now we can evaluate the provided answer options:

A. 3 centimeters → Not possible (not greater than 4)
B. 4 centimeters → Not possible (not greater than 4)
C. 5 centimeters → Possible (greater than 4 and less than 20)
D. 8 centimeters → Possible (greater than 4 and less than 20)
E. 20 centimeters → Not possible (not less than 20)

Therefore, the possible measures of the third side of the triangle are:

**C. 5 centimeters**
**D. 8 centimeters**