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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?

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

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

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

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

User Corentor
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1 Answer

3 votes

Answer:

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

Explanation:

Let be
a and
b rational numbers whose forms are, respectively:


a = (m)/(n) and
b = (q)/(r),
m,
n,
q,
r
\in \mathbb{Z}

We proceed to demonstrate if the Pythagorean sum of two rational numbers leads to a rational number.

1)
a = (m)/(n),
b = (q)/(r),
m,
n,
q,
r
\in \mathbb{Z} Given.

2)
c^(2) = a^(2)+b^(2) Given.

3)
c^(2) = \left((m)/(n) \right)^(2)+\left((q)/(r) \right)^(2) 1) in 2)

4)
c^(2) = (m\cdot n^(-1))^(2)+(q\cdot r^(-1))^(2) Definition of division.

5)
c^(2) = [m^(2)\cdot (n^(-1))^(2)]+[q^(2)\cdot (r^(-1))^(2)]
(a\cdot b)^(c) = a^(c)\cdot b^(c)

6)
c^(2) = m^(2)\cdot n^(-2)+q^(2)\cdot r^(-2)
(a^(b))^(c) = a^(b\cdot c)

7)
c^(2) = [m^(2)\cdot (n^(2))^(-1)]+[q^(2)\cdot (r^(2))^(-1)]
(a^(b))^(c) = a^(b\cdot c)

8)
c^(2) = (m^(2))/(n^(2))+(q^(2))/(r^(2)) Definition of division.

9)
c^(2) = (m^(2)\cdot r^(2)+q^(2)\cdot n^(2))/(n^(2)\cdot r^(2))
(x)/(y) +(w)/(z) = (x\cdot z+w\cdot y)/(y\cdot z)

10)
c^(2) = ((m\cdot r)^(2)+(q\cdot n)^(2))/((n\cdot r)^(n))
(a\cdot b)^(c) = a^(c)\cdot b^(c)/Result.

If
c is a rational number, then
(m\cdot r)^(2)+(q\cdot n)^(2) = a^(2), so that
a \in \mathbb{Z}. Which means that
c will be rational number if and only if the square root of
(m\cdot r)^(2)+(q\cdot n)^(2) is an entire number.

Which means that correct answer is B.

User Jon Story
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