explain in your own words the difference between how the Pythagorean Theorem is used and how the converse of the Pythagorean Theorem is used.
The Pythagorean Theorem is a mathematical principle that describes the relationship between the sides of a right triangle. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
The Pythagorean Theorem is primarily used to find the length of a missing side in a right triangle. To use the Pythagorean Theorem, you need to know the lengths of two sides of the triangle and you can solve for the length of the missing side.
On the other hand, the converse of the Pythagorean Theorem is a statement that reverses the roles of the sides. It states that if the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.
The converse of the Pythagorean Theorem can be used to determine if a given set of three side lengths forms a right triangle. To apply the converse, you need to compare the squares of the sides. If one side's square is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. If not, then it is not a right triangle.
So, in summary, the Pythagorean Theorem is used to find the length of a missing side in a right triangle, while the converse of the Pythagorean Theorem is used to verify if a triangle is a right triangle based on the given side lengths.
Now wait a minute, you are asked to put this in your own words, not my words.
The theorem is:
If it is a right triangle then the sum of the squares of the two legs is equal to the hypotenuse squared.
How do you use that fact? For example if it is a right triangle with legs of 3 and 4, what is the length of the hypotenuse?
Then the converse:
If the square of the longest side is equal to the sum of the squares of the other two sides, then the angle opposite the longest side is a right angle.
So if you have a triangle with sides of 3, 4, and 5, how big is the angle opposite the side of length 5? What is an altitude of the triangle and therefore what is its area?