classifying triangles using the converse of the Pythagorean

Front

What is the Pythagorean theorem?

Back

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). It is expressed as a² + b² = c².

Front

What is the converse of the Pythagorean theorem?

Back

The converse of the Pythagorean theorem states that if in a triangle, the square of the length of one side is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.

Front

How can you determine if a triangle is acute using side lengths?

Back

A triangle is acute if the square of the longest side is less than the sum of the squares of the other two sides (c² < a² + b²).

Front

How can you determine if a triangle is obtuse using side lengths?

Back

A triangle is obtuse if the square of the longest side is greater than the sum of the squares of the other two sides (c² > a² + b²).

Front

What are the side lengths of a right triangle?

Back

In a right triangle, the side lengths must satisfy the Pythagorean theorem, meaning a² + b² = c², where c is the hypotenuse.

Front

What is an example of a set of side lengths that form a right triangle?

Back

An example is the side lengths 9, 12, and 15, which satisfy the Pythagorean theorem: 9² + 12² = 15².

Front

What is an example of a set of side lengths that form an obtuse triangle?

Back

An example is the side lengths 6, 8, and 12, which satisfy the condition for obtuse triangles: 12² > 6² + 8².

Front

What is an example of a set of side lengths that form an acute triangle?

Back

An example is the side lengths 2, 12, and 13, which satisfy the condition for acute triangles: 13² < 2² + 12².

Front

What is the significance of the longest side in triangle classification?

Back

The longest side of a triangle is crucial for determining the type of triangle (acute, right, or obtuse) based on the relationship between its square and the sum of the squares of the other two sides.

Front

What is the relationship between the angles and the type of triangle?

Back

In a right triangle, one angle is 90 degrees; in an acute triangle, all angles are less than 90 degrees; and in an obtuse triangle, one angle is greater than 90 degrees.