Allen is given three lengths of rope: 4 feet, 8 feet, and 15 feet. Can Allen form a triangle with side lengths of 4 feet, 8 feet, and 15 feet using these three pieces of rope, why or why not?
Yes, Allen can form a triangle with side lengths of 4 feet, 8 feet, and 15 feet using these three pieces of rope. This is because of the triangle inequality theorem which states that the sum of any two sides of a triangle must be greater than the third side.
In this case, the sum of the two shortest sides (4 feet and 8 feet) is 12 feet, which is less than the longest side (15 feet). Therefore, the three rope lengths can form a triangle.
By the Triangle Inequality Theorem, If two sides of a triangle have lengths of 3 and 7, what are all the possible lengths of the third side?
The Triangle Inequality Theorem states that the length of the third side of a triangle must be less than the sum of the lengths of the other two sides and greater than the difference between the lengths of the other two sides.
So, if the two sides of a triangle have lengths of 3 and 7, the possible lengths of the third side can be found by:
- The length of the third side is less than the sum of the other two sides: 3 + 7 = 10, so the third side must be less than 10.
- The length of the third side is greater than the difference between the other two sides: 7 - 3 = 4, so the third side must be greater than 4.
Therefore, the possible lengths of the third side of the triangle are any number between 4 and 10 (not including 4 and 10 themselves).
To determine if Allen can form a triangle with side lengths of 4 feet, 8 feet, and 15 feet using the given ropes, we need to check if this set of side lengths satisfies the triangle inequality theorem.
According to the triangle inequality theorem, in a triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. Mathematically, for a triangle with side lengths a, b, and c, the following conditions should hold true:
a + b > c
a + c > b
b + c > a
Let's check these conditions with the given set of side lengths:
4 + 8 > 15 ? (12 > 15) - False
4 + 15 > 8 ? (19 > 8) - True
8 + 15 > 4 ? (23 > 4) - True
Since the first condition (4 + 8 > 15) is false, Allen cannot form a triangle with side lengths of 4 feet, 8 feet, and 15 feet using these three pieces of rope.
To arrive at this conclusion, we used the triangle inequality theorem, which states that the sum of any two sides in a triangle must be greater than the length of the remaining side.