Question

1. in rhombus abcd, which of the following conditions, when added, means that the rhombus is in fact a square? a. ab = ad b. ab⊥bc c. ac⊥bd d. ac bisects ∠bad
2. which of the following statements must be correct? a. diagonals of trapezoids are equal. b. diagonals of rhombi are equal. c. diagonals of rectangles are perpendicular. d. diagonals of parallelograms bisect each other.
3. given that the mid - segment of a trapezoid is 10cm and the height is 5cm, then the area of this trapezoid is ______ cm².

Explanation:

Step1: Recall properties of quadrilaterals

For the first question, a rhombus is a parallelogram with all - sides equal. A square is a rhombus with one right - angle. In a rhombus \(ABCD\), if \(AB = AD\), it is still a rhombus. If \(AB\perp BC\), a rhombus with one right - angle is a square. If \(AC\perp BD\), it is a rhombus. If \(AC\) bisects \(\angle BAD\), it is a rhombus.
For the second question:

• Diagonals of trapezoids are not equal in general.
• Diagonals of rhombi are perpendicular but not equal in general.
• Diagonals of rectangles are equal but not perpendicular in general.
• Diagonals of parallelograms bisect each other.

For the third question, recall the area formula of a trapezoid \(A = mh\), where \(m\) is the mid - segment and \(h\) is the height.

Step2: Solve each question

1. In a rhombus \(ABCD\), the condition \(AB\perp BC\) makes the rhombus a square.
2. The correct statement is that the diagonals of parallelograms bisect each other.
3. Given \(m = 10\mathrm{cm}\) and \(h=5\mathrm{cm}\), using the formula \(A = mh\), we have \(A=10\times5 = 50\mathrm{cm}^2\).

Answer:

1. B
2. D
3. 50