5.1 Triangle Inequality Advanced

Front

What is the Triangle Inequality Theorem?

Back

The Triangle Inequality Theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

Front

Given two sides of a triangle, how do you determine the range of possible lengths for the third side?

Back

To find the range of possible lengths for the third side, 'x', given two sides 'a' and 'b', the inequalities are: |a - b| < x < a + b.

Front

If two sides of a triangle are 4 and 12, what is the range of values for the third side?

Back

The range of values for the third side is between 8 and 16.

Front

What is the range of possible lengths for the third side if the two sides are 8 and 15?

Back

The range of possible lengths for the third side is between 7 and 23.

Front

For two equal sides of a triangle measuring 12, what is the range for the third side?

Back

The range for the third side is 0 < x < 24.

Front

What is a possible length for the third side of a triangle with sides measuring 13 and 4?

Back

A possible length for the third side is 12.

Front

How can you determine if a triangle with sides 15, 12, and 9 exists?

Back

To determine if a triangle exists, check if the sum of the lengths of any two sides is greater than the length of the third side. In this case, 15 + 12 > 9, 15 + 9 > 12, and 12 + 9 > 15, so the triangle exists.

Front

What is the significance of the Triangle Inequality Theorem in geometry?

Back

The Triangle Inequality Theorem is significant because it provides a necessary condition for the existence of a triangle given the lengths of its sides.

Front

Can a triangle have sides measuring 1, 2, and 3? Why or why not?

Back

No, a triangle cannot have sides measuring 1, 2, and 3 because 1 + 2 is not greater than 3, violating the Triangle Inequality Theorem.

Front

What happens if the sum of two sides of a triangle equals the third side?

Back

If the sum of two sides equals the third side, the points are collinear, and a triangle cannot be formed.