Determine if the given side lengths could be used to form a unique triangle, many different triangles, or no triangles.
12 ft, 7 ft, 14 ft
A.
unique triangle
B.
many different triangles
C.
no triangles
Hey! Can you guys help me out with this? I don't really know how to do it.
As long as the largest side is less than the sum of the other two sides, they can form a triangle. In fact, A.
This is handy in construction, because triangular braces cannot be squashed easily. Rectangles, though, can be skewed into parallelograms without changing the side lengths.
thank you so so so much oobleck!!
Sure! To determine if the given side lengths can form a unique triangle, many different triangles, or no triangles, we can use the triangle inequality theorem.
According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's check if this condition is satisfied for the given side lengths (12 ft, 7 ft, and 14 ft):
1. The sum of the lengths of the first two sides (12 ft and 7 ft) is 12 + 7 = 19 ft. Since 19 ft is greater than the length of the third side (14 ft), this condition is satisfied.
2. The sum of the lengths of the first and third sides (12 ft and 14 ft) is 12 + 14 = 26 ft. Since 26 ft is greater than the length of the second side (7 ft), this condition is satisfied.
3. The sum of the lengths of the second and third sides (7 ft and 14 ft) is 7 + 14 = 21 ft. Since 21 ft is greater than the length of the first side (12 ft), this condition is satisfied.
Since all three conditions are satisfied, it is possible to form a triangle with side lengths of 12 ft, 7 ft, and 14 ft.
However, to determine the uniqueness of the triangle, we need additional information like the angles. With only the side lengths given, we cannot determine if it will be a unique triangle or many different triangles. Therefore, the correct answer is:
B. many different triangles
Absolutely! I'd be happy to help you solve this problem.
To determine if the given side lengths can form a unique triangle, many different triangles, or no triangles, we need to apply the triangle inequality theorem. According to the theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
For the given side lengths of 12 ft, 7 ft, and 14 ft, let's test this condition for each pair of sides:
1. 12 ft + 7 ft = 19 ft > 14 ft
2. 12 ft + 14 ft = 26 ft > 7 ft
3. 7 ft + 14 ft = 21 ft > 12 ft
Since the sum of the two shorter sides is always greater than the longest side in all three cases, we can conclude that a triangle can be formed using these side lengths.
Now, let's determine if it can form a unique triangle, many different triangles, or no triangles. To do this, we can check if all pairs of sides are unequal. If they are, then we have a unique triangle. If any two sides are equal, then many different triangles can be formed. If all three sides are equal, then no triangles can be formed.
1. 12 ft ≠ 7 ft
2. 12 ft ≠ 14 ft
3. 7 ft ≠ 14 ft
Since all three pairs of sides are unequal, we can conclude that a unique triangle can be formed using the given side lengths.
Therefore, the answer is A. unique triangle.