Inequalities in Triangles

Front

What is an inequality in a triangle?

Back

An inequality in a triangle refers to the relationship between the lengths of the sides and the measures of the angles. Specifically, the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle.

Front

What is the Triangle Inequality Theorem?

Back

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

Front

If two sides of a triangle are 8 and 10, what is the range of possible lengths for the third side?

Back

The length of the third side must be greater than 2 (10 - 8) and less than 18 (10 + 8). So, the range is (2, 18).

Front

How do you determine the smallest angle in a triangle?

Back

The smallest angle in a triangle is opposite the shortest side.

Front

If a triangle has sides of lengths 5, 12, and 13, is it a right triangle?

Back

Yes, it is a right triangle because 5² + 12² = 13² (25 + 144 = 169).

Front

What is the relationship between the angles and sides of a triangle?

Back

In a triangle, the larger the angle, the longer the side opposite that angle.

Front

If the angles of a triangle are 30°, 60°, and 90°, what are the ratios of the sides?

Back

The ratios of the sides opposite these angles are 1:√3:2.

Front

What can be concluded if one angle of a triangle is greater than 90°?

Back

If one angle of a triangle is greater than 90°, the triangle is classified as an obtuse triangle.

Front

What is the maximum length of the third side if two sides of a triangle are 7 and 10?

Back

The maximum length of the third side can be 16 (7 + 10 - 1).

Front

If a triangle has sides of lengths 3, 4, and x, what is the range of possible values for x?

Back

The range for x is (1, 7) because 3 + 4 > x, 3 + x > 4, and 4 + x > 3.