Sides of a quadrilateral are all positive integers.How many possible values the fourth side have if three of its sides are 5cm,10cm,20cm?
To find the possible values for the fourth side of the quadrilateral, we need to apply the triangle inequality which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
In this case, we are given that three sides of the quadrilateral are 5 cm, 10 cm, and 20 cm. Let's call the fourth side x cm.
According to the triangle inequality, we can write the following inequalities:
1. 5 cm + 10 cm > x cm
2. 5 cm + 20 cm > x cm
3. 10 cm + 20 cm > x cm
Simplifying these inequalities:
1. 15 cm > x cm
2. 25 cm > x cm
3. 30 cm > x cm
Since all sides are positive integers, we need to find the largest integer value that satisfies all three inequalities.
From the first inequality, we know that x must be less than 15 cm. From the second inequality, x must be less than 25 cm. From the third inequality, x must be less than 30 cm.
Therefore, the largest possible value for the fourth side (x) of the quadrilateral is the largest integer that is less than all three of these upper bounds, which is 14 cm.
Hence, the fourth side of the quadrilateral can have a maximum value of 14 cm.
Hint:
The three sides cannot even form a triangle, because 20>5+10.
Let X=length of fourth side, then
Since 20-(5+10)=5,
the fourth side must be greater than 5, or X>5
Similarly, the fourth side cannot be longer than the sum of the other three sides, otherwise the quadrilateral will never close.
This means that X<5+10+20=35
Summing up, we have side X such that
5<X<35, given that X must be an integer.
It's up to you to count the number of possible values of X.