Is it possible to construct a quadrilateral ABCD in which AB = 1.5 cm, AD = 3 cm, CD = 2 cm, AC = 6.5 cm, BD = 5 cm?

To determine if it is possible to construct a quadrilateral with the given side lengths, we need to apply the triangle inequality theorem to each combination of sides. The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Triangle Inequality Theorem:
1. The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
2. If the sum of the lengths of any two sides of a triangle is equal to the length of the third side, then the triangle is degenerate (a straight line).

Let's analyze the given side lengths:

1. AB + CD > AC:
1.5 cm + 2 cm = 3.5 cm (which is less than 6.5 cm)
Therefore, AB + CD < ac,="" satisfying="" the="" triangle="" />

2. AD + BC > AC:
3 cm + BC > 6.5 cm
BC > 6.5 cm - 3 cm
BC > 3.5 cm

Since we don't have the length of BC, we cannot determine whether AD + BC > AC or not. So, let's continue analyzing the side lengths.

3. AC + BD > AD:
6.5 cm + 5 cm = 11.5 cm (which is greater than 3 cm)
Therefore, AC + BD > AD, satisfying the triangle inequality.

4. AB + BD > AD:
1.5 cm + 5 cm = 6.5 cm (which is equal to 3 cm)
According to the triangle inequality theorem, when the sum of two sides is equal to the length of the third side, the triangle formed is degenerate (a straight line). This means that AB and BD must be collinear, which is not possible in a quadrilateral.

Based on the analysis above, it is not possible to construct a quadrilateral ABCD with the given side lengths of AB = 1.5 cm, AD = 3 cm, CD = 2 cm, AC = 6.5 cm, and BD = 5 cm.

No