Question

The Pythagorean theorem states that the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse by the formula a2 + b2 = c2.

Answer 1

The first statement comes about because 
a could be (say) 5
b could be (say) 6

Then c would be square root of 25 + 36 = 61 which is irrational.

I'm not sure of the wording of the second one. It reads like (5 + 6)^2 = 121 the two rationals added together make a rational number in my example. So I think that part is wrong. the second part is correct. For example sqrt(5) + sqrt(6) doesn't produce anything that is rational.

It doesn't have to. pi^2 + [cube root(10)]^2 does not produce anything rational and that is certainly not a perfect square. My first example shows that as well

I think the last one might be true.

Answer 2

Question with four option

The Pythagorean theorem states that the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse by the formula, a² + b² = c².If a is a rational number and b is a rational number, why could c be an irrational number?

The square of rational numbers is irrational, and sum of two irrational numbers is irrational.

The product of two rational numbers is rational, and the sum of two rational numbers is irrational.

The left side of the equation will result in a rational number, which is a perfect square.

The left side of the equation will result in a rational number, which could be a non-perfect square.

Answer with Explanation

The Pythagorean theorem states that

          Sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse.

   ⇒   a² + b² = c²

It has been asked that, a is a rational number and b is a rational number ,then why c is an Irrational number.

Suppose, a=3 ,b=4 , the set of number forming Pythagorean triplet

Then, ⇒ a² + b²

  =3²+4²

  = 9 +16

 =25

  = 5²

Here, c=5 is a Rational Number.

Now, take, a=2, b=3,the set of number not forming Pythagorean triplet

a²+b²

=2²+3²

=4 +9

=13

Now,  a² + b² = c²

⇒ c²=13

$\rightarrow c^2=13\\\\c=\pm \sqrt{13}$

But side of a triangle can't be negative.

So,Length of  Hypotenuse =√13

It totally depends on which two numbers you are choosing, A set of numbers forming Pythagorean Triplet or any other set of numbers.

   → If you choose any other set of numbers which is not a Pythagorean triplet  , the Hypotenuse will be an irrational number.

Option D:→ The left side of the equation will result in a rational number, which could be a non-perfect square.