Question

4 which of the following statements is true? a. a quadrilateral with equal diagonals is an isosceles trapezoid. b. a trapezoid with two equal adjacent interior angles is an isosceles trapezoid. c. a quadrilateral with one pair of opposite sides parallel and the other pair of opposite sides equal is an isosceles trapezoid. d. a quadrilateral obtained by cutting two legs of an isosceles triangle with a line parallel to the base is an isosceles trapezoid. 5 given that the median of a trapezoid is 6cm and the altitude is 5cm, then the area of the trapezoid is ______ cm². 6 which of the following statements is correct? a. the diagonal of a rectangle bisects one pair of opposite angles. b. the diagonals of a parallelogram are equal. c. the diagonals of a trapezoid are perpendicular to each other. d. the diagonals of a rhombus perpendicular bisect each other.

Explanation:

4.

• Option A: A quadrilateral with equal diagonals is not necessarily an isosceles trapezoid, it could be a rectangle etc.
• Option B: A trapezoid with two equal adjacent - interior angles is not necessarily an isosceles trapezoid.
• Option C: A quadrilateral with one pair of opposite sides parallel and the other pair of opposite sides equal is not always an isosceles trapezoid, it could be a parallelogram.
• Option D: When we cut two legs of an isosceles triangle with a line parallel to the base, the resulting quadrilateral has one - pair of parallel sides (parallel to the base of the original triangle) and non - parallel sides are equal (due to the similarity of triangles and the property of the isosceles triangle), so it is an isosceles trapezoid.

Step1: Recall the area formula of a trapezoid

The area formula of a trapezoid is $A =$ median $\times$ altitude.

Step2: Substitute the given values

Given that the median $m = 6$ cm and the altitude $h=5$ cm. Substitute into the formula $A=m\times h$.
$A = 6\times5$

• Option A: The diagonal of a rectangle does not bisect one pair of opposite angles. Only the diagonals of a square (a special rectangle) bisect the opposite angles.
• Option B: The diagonals of a general parallelogram are not equal. The diagonals of a rectangle (a special parallelogram) are equal.
• Option C: The diagonals of a general trapezoid are not perpendicular to each other.
• Option D: The diagonals of a rhombus are perpendicular bisectors of each other, which is a well - known property of a rhombus.

Answer:

D. A quadrilateral obtained by cutting two legs of an isosceles triangle with a line parallel to the base is an isosceles trapezoid.

5.