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. $overline{ab}perpoverline{bc}$ c. $overline{ac}perpoverline{bd}$ d. $overline{ac}$ bisects $angle bad$ 6. 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. 7. given that the mid - segment of a trapezoid 10cm and the height is 5cm, then the area of this trapezoid is ______ $cm^{2}$.

Explanation:

Step1: Recall properties of rhombus and square

A rhombus has all - sides equal. A square is a rhombus with one right - angle.

Step2: Analyze option for question 5

In a rhombus \(ABCD\), if \(\overline{AB}\perp\overline{BC}\), then one of the angles of the rhombus is a right - angle. Since a rhombus with a right - angle is a square, option B is correct for question 5.

Step3: Recall properties of quadrilaterals for question 6

The diagonals of a parallelogram bisect each other. Trapezoids generally have non - equal diagonals, rhombi have perpendicular but not necessarily equal diagonals, and rectangles have equal but not necessarily perpendicular diagonals. So, option D is correct for question 6.

Step4: Use trapezoid area formula for question 7

The area formula of a trapezoid is \(A = mh\), where \(m\) is the mid - segment and \(h\) is the height. Given \(m = 10\mathrm{cm}\) and \(h=5\mathrm{cm}\), then \(A=10\times5 = 50\mathrm{cm}^2\).

Answer:

1. B. \(\overline{AB}\perp\overline{BC}\)
2. D. Diagonals of parallelograms bisect each other.
3. 50