Math Task

Liceo de Cagayan University

High School Department

SY: 2014-2015

Performance Task

In

Mathematics

Submitted by:

Clyde Sabuero

9-Einstein

Submitted to:

Mrs. Cherly Yabo

Math Teacher

Instructions: Research on at least three different proofs of the Pythagorean Theorem. Present these in details with their appropriate illustrations.

#1

Bhaskara's First Proof

Bhaskara's proof is also a dissection proof. It is similar to the proof provided by Pythagoras. Bhaskara was born in India. He was one of the most important Hindu mathematicians of the second century AD. He used the following diagrams in proving the Pythagorean Theorem. 

In the above diagrams, the blue triangles are all congruent and the yellow squares are congruent. First we need to find the area of the big square two different ways. First let's find the area using the area formula for a square. 

Thus, A=c^2.

Now, let us find the area by finding the area of each of the components and then sum the areas.

Area of the blue triangles = 4(1/2)ab

Area of the yellow square = (b-a)^2

Area of the big square = 4(1/2)ab + (b-a)^2

= 2ab + b^2 - 2ab + a^2

= b^2 + a^2

Since, the square has the same area no matter how you find it

A = c^2 = a^2 + b^2,

concluding the proof.

#2

Bhaskara's Second Proof of the Pythagorean Theorem

In this proof, Bhaskara began with a right triangle and then he drew an altitude on the hypotenuse. From here, he used the properties of similarity to prove the theorem.

Now prove that triangles ABC and CBE are similar.

It follows from the AA postulate that triangle ABC is similar to triangle CBE, since angle B is congruent to angle B and angle C is congruent to angle E. Thus, since internal ratios are equal s/a=a/c.

Multiplying both sides by ac we get

sc=a^2.

Now show that triangles ABC and ACE are similar.

As before, it follows from the AA postulate that these two triangles are similar. Angle A is congruent to angle A and angle C is congruent to angle...