A triangle is made of wood sticks of lengths 8, 15 and 17 inches joined end-to-end. Pieces of the same integral length are cut from each of the sticks so that the three remaining pieces can no longer form a triangle. How many inches are in the length of the smallest piece that can be cut from each of the three sticks to make this happen?
clearly, when the longest side is the sum of the other two sides, the triangle has flattened into a line. So,
17-x = 8-x + 15-x
17-x = 23-2x
x = 6
cutting off 6" leaves the lengths of 2,9,11, a flat triangle.
Our current triangle lengths are 8, 15, and 17. Let us say that x is the length of the piece that we cut from each of the three sticks. Then, our lengths will be 8 - x, 15 - x, and 17 - x. These lengths will no longer form a triangle when the two shorter lengths added together is shorter than or equal to the longest length. In other words, (8 - x) + (15 - x) ≤ (17 - x). Then, we have 23 - 2x ≤ 17 - x, so 6 ≤ x. Therefore, the length of the smallest piece that can be cut from each of the three sticks is *6 inches*.
To find the length of the smallest piece that can be cut from each of the three sticks, we need to consider the conditions for a triangle to be formed. 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.
In this case, the original lengths of the three wood sticks are 8, 15, and 17 inches. Now, we need to find the smallest piece that can be cut from each stick such that the remaining pieces can no longer form a triangle.
To do this, we can start by considering the longest side of the original triangle, which is 17 inches. If we remove the smallest piece from this side, the remaining two sides need to satisfy the triangle inequality theorem.
Let's assume the length of the smallest piece cut from the longest side is x inches. Therefore, the remaining two sides would be (17 - x) and 15 inches.
For a triangle to not be formed, the sum of the two remaining sides must be less than or equal to the length of the original longest side:
(17 - x) + 15 ≤ 17
Simplifying the inequality:
32 - x ≤ 17
Subtracting 32 from both sides:
-x ≤ -15
Since x is a positive length, we can multiply the inequality by -1 to switch the direction of the inequality sign:
x ≥ 15
Thus, the smallest piece that can be cut from each of the three sticks to make the remaining pieces unable to form a triangle must have a length of at least 15 inches.
the sum of the two smaller sides must be greater than the largest side to have a triangle
so if we cut off x units from each stick
8-x + 15-x = 17-x
-x = -6
x = 6
so if we cut off 6 inches, our sticks will form a straight line, so
we can cut off 5 inches from each and still get a triangle.