Determine if the given information would form a
unique triangle, many different triangles, or no triangles.
9 inches, 7 inches, 18 inches
Mark sides as:
a = 9 , b = 7 , c = 18
Apply triangle inequality theorem:
a + b > c
b + c > a
a + c > b
This rule must be satisfied for ALL three conditions of the sides.
First condition:
a + b > c
9 + 7 > 18
16 > 18
This is obviously not true, so sides:
a = 9 , b = 7 , c = 18
cannot be sides of any triangle.
if the sides are a,b,c in increasing order, then you must have
b-a < c < b+a
for your numbers, you need
9-7 < 18 < 9+7
2 < 18 < 16
which is false, so no triangles.
Think of it. If you lay down the sides 7 and 9, they are not long enough to even cover the base of length 18, much less fold up to form a triangle.
To determine if the given information would form a unique triangle, many different triangles, or no triangle, we can apply the Triangle Inequality Theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must always be greater than the length of the third side.
Let's analyze the given information:
Side length A: 9 inches
Side length B: 7 inches
Side length C: 18 inches
Now, we need to check if the sum of any two sides is greater than the length of the third side:
1. A + B > C: 9 + 7 > 18. This is false since 9 + 7 = 16, which is not greater than 18.
2. A + C > B: 9 + 18 > 7. This is true since 9 + 18 = 27, which is greater than 7.
3. B + C > A: 7 + 18 > 9. This is true since 7 + 18 = 25, which is greater than 9.
Since at least one of the inequalities (2 and 3) is true, the given side lengths would form a triangle. However, it is important to note that only one triangle can be formed using these specific side lengths.