Question

1. as shown in the figure, in a square abcd whose side length is 2, m is the mid - point of ad. extend md until reaching point e such that me = mc. create a square defg with one of its sides being de. point g is on cd. then dg = ______. a. $sqrt{3}-1$ b. $3 - sqrt{5}$ c. $sqrt{5}+1$ d. $sqrt{5}-1$ 2. which of the following statements is incorrect? a. a rectangle with one pair of equal adjacent sides is a square. b. a rhombus with equal diagonals is a square. c. a rectangle whose diagonals are perpendicular to each other is a square. d. a parallelogram with one angle being right angle is a square.

Explanation:

Step1: Calculate MC and MD lengths

In square \(ABCD\) with side - length \(a = 2\), \(M\) is the mid - point of \(AD\), so \(AM=MD = 1\), \(DC = 2\).
By the Pythagorean theorem in right - triangle \(MDC\), \(MC=\sqrt{MD^{2}+DC^{2}}=\sqrt{1^{2}+2^{2}}=\sqrt{1 + 4}=\sqrt{5}\).
Since \(ME = MC=\sqrt{5}\), then \(DE=ME - MD=\sqrt{5}-1\).
Because \(DEFG\) is a square, \(DG = DE=\sqrt{5}-1\).

Step2: Analyze the second question

Option A:

A rectangle has all angles equal to \(90^{\circ}\). If one pair of adjacent sides is equal, it satisfies the definition of a square.

Option B:

A rhombus has all sides equal. If its diagonals are equal, it is a square according to the properties of special quadrilaterals.

Option C:

A rectangle has all angles equal to \(90^{\circ}\). If its diagonals are perpendicular to each other, it is a square.

Option D:

A parallelogram with one right - angle is a rectangle, not necessarily a square. A square requires all sides to be equal in addition to having right - angles.

Answer:

1. D. \(\sqrt{5}-1\)
2. D. A parallelogram with one angle being right angle is a square.