Question

it is possible that in a parallelogram with side lengths of 4 cm and 7 cm, the length of one of the diagonals is 2 cm? answer: it is . you have only 1 attempt to earn the full credit for this problem.

Explanation:

Step1: Recall triangle - inequality in parallelogram

In a parallelogram, the two - side lengths and a diagonal form a triangle. Let the side lengths of the parallelogram be \(a = 4\mathrm{cm}\) and \(b = 7\mathrm{cm}\), and the diagonal be \(c\). According to the triangle - inequality theorem, for a triangle with side lengths \(x\), \(y\), and \(z\), the following three inequalities must hold: \(x + y>z\), \(x+z > y\), and \(y + z>x\).

Step2: Check the triangle - inequality

Suppose the diagonal \(c = 2\mathrm{cm}\), and the two sides of the triangle formed by the sides of the parallelogram and the diagonal are \(a = 4\mathrm{cm}\) and \(b = 7\mathrm{cm}\). Consider the sum of the lengths of two sides of the triangle formed by the sides of the parallelogram and the diagonal. If we take \(a = 4\mathrm{cm}\) and \(c = 2\mathrm{cm}\), then \(a + c=4 + 2=6\mathrm{cm}\), and \(6\mathrm{cm}<7\mathrm{cm}\), which violates the triangle - inequality theorem (\(a + c>b\)).

Answer:

It is not possible.