# Answer to Question #17032 in Analytic Geometry for Nijwm

Question #17032

The sides of Quadrilaterals ABCD are AB=6cm, BC=18cm,CD=6cm and DA=10cm. If the restriction is that the diagonal BD must be an integer, how many such Quadrilaterals can be formed?

Expert's answer

Consider the triangle BDA. AB = 6, AD = 10.

According to the triangle rule the triangle BDA can be formed only when |AB - AD| < BD < AB + BD.

So, 4 < BD < 16

Consider the triangle BDC. CB = 18, CD = 6.

According to the triangle rule the triangle BDC can be formed only when |CB - CD| < BD < CB + CD.

So, 12 < BD < 24

Quadrilateral ABCD can be formed when both triangles BDA and BDC can be formed.

So, 12 < BD < 16.

BD must be integer, so BD can be equal to 13, 14 and 15 – 3 different values in total.

So, 3 different Quadrilaterals can be formed.

According to the triangle rule the triangle BDA can be formed only when |AB - AD| < BD < AB + BD.

So, 4 < BD < 16

Consider the triangle BDC. CB = 18, CD = 6.

According to the triangle rule the triangle BDC can be formed only when |CB - CD| < BD < CB + CD.

So, 12 < BD < 24

Quadrilateral ABCD can be formed when both triangles BDA and BDC can be formed.

So, 12 < BD < 16.

BD must be integer, so BD can be equal to 13, 14 and 15 – 3 different values in total.

So, 3 different Quadrilaterals can be formed.

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