Question

The perimeter of a triangle is $2004$. One side of the triangle is $21$ times the other. The shortest side is of integral length. If length of one side of the triangle, in every possible case, is $x,$ then $x = $
A) $47$ or $48$
B) $46$ or $47$
C) $45$ or $46$
D) $45$ or $48$

Answer 1

Hint: To solve this problem, we will use the formula for the perimeter of a triangle. We will use different properties of a triangle to form inequalities and thus get the required answers.

Complete step-by-step answer:

We have been given the perimeter of the triangle as $2004$ .
Also, it is given in the problem that one side of the triangle is $21$ times the other.
Let’s assume a triangle $\Delta ABC$ of side lengths
\[AB{\text{ }} = {\text{ }}c\] [Assumed to be the largest side]
\[AC{\text{ }} = {\text{ }}b\]
\[BC{\text{ }} = {\text{ }}a\] [Assumed to be the shortest side]
Let us assume that the length of the largest side is $21$ times that of the shortest side.
Thus, mathematically,
$c\, = \,21a$ ………. Eq I
Let’s take the case when the shortest side, i.e.,
$a = x$ ………. Eq II
Also we know that
$primeter\,\,of\,\,triangle\,\, = \,\,sum\,\,of\,\,three\,\,sides$
$ \Rightarrow \,\,a + b + c\,\, = \,\,perimeter$
$ \Rightarrow \,\,x + b + 21x\,\, = \,\,2004$ [Using Equations I and II]
$ \Rightarrow \,\,b\,\, = \,\,2004 - 22x$ ……. Eq III
Thus the three sides of the triangle are
$a = x$
$b = 2004 - 22x$
$c = 21x$
We know, that the sum of two sides of a triangle is always greater than the third side:
$ \Rightarrow \,\,a + c > b$
$ \Rightarrow \,\,x + 21x > 2004 - 22x$
$ \Rightarrow \,\,44x > 2004$
$ \Rightarrow \,\,x > \dfrac{{2004}}{{44}}$
$ \Rightarrow \,\,x > 45.\bar 5\bar 4$ ……. Eq IV
Also, we know that difference of two sides of a triangle is always less than the third side:
$ \Rightarrow \,c - a < b$
$ \Rightarrow \,21x - x < 2004 - 22x$
$ \Rightarrow \,20x < 2004 - 22x$
$ \Rightarrow \,42x < 2004$
$ \Rightarrow \,x < 47.7143$ ……. Eq V
From equations IV and V, it can be concluded that:
$45.\bar 5\bar 4 < x < 47.7143$
Thus the value of $x$ cannot be 45 or 48 if we insist on integers.
It can be either 46 or 47.
Cross-verification $x = 46$ :
$a = x = 46$
$b = 2004 - 22x = 2004 - 22 \times 46 = 992$
\[c = 21x = 21 \times 46 = 966\]
$ \Rightarrow perimeter = a + b + c = \,46 + 992 + 966 = 2004$
Cross-verification $x = 47$:
$a = x = 47$
$b = 2004 - 22x = 2004 - 22 \times 47 = 970$
\[c = 21x = 21 \times 47 = 987\]
$ \Rightarrow perimeter = a + b + c = \,47 + 970 + 987 = 2004$

Therefore, option (B) is correct.

Note: Integral side lengths don’t have any resemblance to integration and calculus. It simply means that the length must not be a decimal or fraction, must be an integer.