The lengths of all four sides of a quadrilateral are integer valued. If three of its sides are of length $1$ cm, $2$ cm and $4$ cm, then the total number of possible lengths of the fourth side is

We have a quadrilateral with three known sides: 1 cm, 2 cm, and 4 cm. We need to find how many possible integer lengths the fourth side can have.

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For any quadrilateral to exist, there's an important rule we must follow:

The sum of any three sides must be greater than the fourth side.

This is similar to the triangle inequality, but extended to quadrilaterals. Think of it this way: if you try to form a quadrilateral with four sticks, you can't make it work if one side is too long compared to the others.

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Let's call the fourth side x cm. We need to check all possible combinations:

Three known sides vs. fourth side:

Sum of three known sides > Fourth side

$1 + 2 + 4 > x$

$7 > x$

$x < 7$

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Two known sides + fourth side vs. remaining known side:

We need: $1 + 2 + x > 4$

$3 + x > 4$

$x > 1$

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Check other combinations:

$1 + 4 + x > 2$ gives us $x > -3$ (always true for positive lengths)

$2 + 4 + x > 1$ gives us $x > -5$ (always true for positive lengths)

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From our inequalities:

$x < 7$ (from condition 1)

$x > 1$ (from condition 2)

Combining these: $1 < x < 7$

Since the length must be an integer, the possible values are:

$x = 2, 3, 4, 5, 6$

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The total number of possible lengths of the fourth side is 5.

The possible lengths are: 2 cm, 3 cm, 4 cm, 5 cm, and 6 cm.