The following side lengths, in meters, were given to a carpenter to build a front porch with a triangular design. The carpenter needs to determine which set of lengths will make a triangle to be able to use it in his design. Option 1: Side lengths: 3, 3, 6 Option 2: Side lengths: 6, 8, 10 Option 3: Side lengths: 4, 6, 13
To determine which set of lengths will make a triangle, we can use 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 remaining side.
Let's check each option:
Option 1: Side lengths: 3, 3, 6
The sum of the two shorter sides is 3 + 3 = 6. This is equal to the length of the longest side. Therefore, this set of lengths does not form a valid triangle.
Option 2: Side lengths: 6, 8, 10
The sum of the two shorter sides is 6 + 8 = 14, which is greater than the length of the longest side (10). Therefore, this set of lengths forms a valid triangle.
Option 3: Side lengths: 4, 6, 13
The sum of the two shorter sides is 4 + 6 = 10, which is less than the length of the longest side (13). Therefore, this set of lengths does not form a valid triangle.
Based on the triangle inequality theorem, option 2 with side lengths 6, 8, and 10 is the only valid triangle.