Question

Which of the following is always rational?

• A. The hypotenuse of a right angled triangle having rational base and perpendicular
• B. The area of a triangle​, given the value of the sides are irrational
• C. The volume of a cube, given the value of the sides are rational
• D. The area of rectangle in which the value of one of the sides is irrational

Answer 1

The correct option is C The volume of a cube, given the value of the sides are rational

From Pythagoras theorem, we know that
(hypotenuse) $h^2$ = (perpendicular) $p^2$ + (base) $b^2$.
Thus, $h=(p^2+b^2{)}^{\frac{1}{2}}.$
Since hypotenuse is square root, it is very much possible that it is irrational as square root of nonperfect squares is irrational. For e.g. If p = 3 and b = 7 then h = $\sqrt{5}8$ which is irrational.
(b) The area of a triangle = $\frac{1}{2}\times b\times h$. If either of b or h is irrational then area becomes irrational.
(c) Volume = (side of cube) $a^3$ which is always rational if the value of `a' is kept rational. Thus, volume of cube is always rational.
(d) Area of rectangle = (length) l $\times$ (base) b. If either of l or b is irrational, area becomes irrational as product of rational and irrational is always irrational.