DAY 77 PROOF OF THE PYTHAGOREAN THEOREM USING

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DAY 77 – PROOF OF THE PYTHAGOREAN THEOREM USING SIMILARITY

DAY 77 – PROOF OF THE PYTHAGOREAN THEOREM USING SIMILARITY

INTRODUCTION The Pythagorean theorem, named in honor of the Greek philosopher, Pythagoras of Samos,

INTRODUCTION The Pythagorean theorem, named in honor of the Greek philosopher, Pythagoras of Samos, is one of the fundamental theorems in geometry. It is also referred to as the Pythagoras’ theorem. It explores the relationship between the squares of the lengths of the three sides of a right triangle. It is possible to prove this theorem using the concept of similar triangles though it can be proved by other ways, including algebra. In this lesson, we will learn how to prove this theorem based on the concept of similarity.

VOCABULARY 1. Hypotenuse The side of a right triangle that is opposite the right

VOCABULARY 1. Hypotenuse The side of a right triangle that is opposite the right angle. 2. Right triangle A triangle that has a right angle as one of its interior angles.

PYTHAGOREAN THEOREM The Pythagorean theorem states that, in a right triangle, the square of

PYTHAGOREAN THEOREM The Pythagorean theorem states that, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. A C B

THE CONVERSE OF PYTHAGORAS’ THEOREM The converse of the Pythagoras’ theorem is equally important:

THE CONVERSE OF PYTHAGORAS’ THEOREM The converse of the Pythagoras’ theorem is equally important: It states that if the sum of the squares of the lengths of two sides of a triangle equals the square of the length of the hypotenuse, then the triangle is a right triangle.

PROOF OF THE PYTHAGOREAN THEOREM BASED ON SIMILARITY The proof of theorem is based

PROOF OF THE PYTHAGOREAN THEOREM BASED ON SIMILARITY The proof of theorem is based on the proportionality of the sides of similar triangles and the altitudes of these triangles. We will base our proof on AA similarity criterion for similar triangles, that is, if two corresponding angles of two triangles are congruent, then the two triangles are similar. We have to recall that in similar triangles corresponding angles are congruent and the corresponding sides are proportional.

P Q R

P Q R

We will also use another theorem without proof to prove the Pythagoras theorem. The

We will also use another theorem without proof to prove the Pythagoras theorem. The theorem states that: If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, then triangles on both sides of the perpendicular are similar to the whole triangle and to each other.

P S Q R

P S Q R

P S Q R

P S Q R

Q P S Q S R R

Q P S Q S R R

C A D B

C A D B

HOMEWORK K N L M

HOMEWORK K N L M

ANSWERS TO HOMEWORK

ANSWERS TO HOMEWORK

THE END

THE END

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