Determine whether the given measure can be the lengths of the sides of a triangle. Write yes or no.
1. 8, 9, 17
2. 23, 26, 50
3. 12.3, 13.9, 25.2
the two shortest sides must sum to GREATER THAN the longest side in order for the triangle to "close up"
Thank you so much!
To determine whether the given measures can be the lengths of the sides of a triangle, we can apply the triangle inequality theorem. According to this theorem, in a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
Let's apply this theorem to each set of measures:
1. For the measures 8, 9, and 17:
- The sum of the lengths of sides 8 and 9 is 17, which is larger than the length of side 17.
- The sum of the lengths of sides 9 and 17 is 26, which is larger than the length of side 8.
- The sum of the lengths of sides 8 and 17 is 25, which is larger than the length of side 9.
Since the sum of the lengths of any two sides in this case is always greater than the length of the third side, we can say that the measures 8, 9, and 17 can be the lengths of the sides of a triangle. Therefore, the answer is "yes."
2. For the measures 23, 26, and 50:
- The sum of the lengths of sides 23 and 26 is 49, which is smaller than the length of side 50.
Since the sum of the lengths of sides 23 and 26 is not greater than the length of side 50, the triangle inequality theorem is not satisfied. Thus, the measures 23, 26, and 50 cannot be the lengths of the sides of a triangle. Therefore, the answer is "no."
3. For the measures 12.3, 13.9, and 25.2:
- The sum of the lengths of sides 12.3 and 13.9 is 26.2, which is larger than the length of side 25.2.
- The sum of the lengths of sides 12.3 and 25.2 is 37.5, which is larger than the length of side 13.9.
- The sum of the lengths of sides 13.9 and 25.2 is 39.1, which is larger than the length of side 12.3.
Since the sum of the lengths of any two sides in this case is always greater than the length of the third side, we can say that the measures 12.3, 13.9, and 25.2 can be the lengths of the sides of a triangle. Therefore, the answer is "yes."
In summary:
1. Yes, the measures 8, 9, and 17 can be the lengths of the sides of a triangle.
2. No, the measures 23, 26, and 50 cannot be the lengths of the sides of a triangle.
3. Yes, the measures 12.3, 13.9, and 25.2 can be the lengths of the sides of a triangle.