Question

1. it is given that the ratio of the lengths of the upper and lower bases of a trapezoid is 1:2. the median divides the trapezoid into two parts. the ratio of the area of the upper part to that of the lower part is ____. 8. if we want to prove that the parallelogram abcd is a square, we need to further prove ____. a. ab = ad and ac = bd b. ab = ad and $overline{ac}perpoverline{bd}$ c. $mangle a=mangle b$ and ac = bd d. $overline{ac}$ and $overline{bd}$ perpendicular bisect each other

Explanation:

Step1: Recall trapezoid - median formula

Let the length of the upper - base of the trapezoid be \(a\), then the length of the lower - base is \(2a\). The length of the median of a trapezoid \(m=\frac{a + 2a}{2}=\frac{3a}{2}\). Let the height of the whole trapezoid be \(h\), and the height of the upper part (formed by the upper - base and the median) be \(h_1\), and the height of the lower part (formed by the median and the lower - base) be \(h_2\). Since the two sub - trapezoids (formed by the median) are similar in a sense, and the ratio of their parallel sides is linear, \(h_1=h_2=\frac{h}{2}\).

Step2: Calculate the area of the upper part

The area formula of a trapezoid is \(A=\frac{(b_1 + b_2)h}{2}\). The area of the upper part \(A_1\) with upper - base \(a\), lower - base \(\frac{3a}{2}\), and height \(\frac{h}{2}\) is \(A_1=\frac{(a+\frac{3a}{2})\frac{h}{2}}{2}=\frac{( \frac{2a + 3a}{2})\frac{h}{2}}{2}=\frac{\frac{5a}{2}\times\frac{h}{2}}{2}=\frac{5ah}{8}\).

Step3: Calculate the area of the lower part

The area of the lower part \(A_2\) with upper - base \(\frac{3a}{2}\), lower - base \(2a\), and height \(\frac{h}{2}\) is \(A_2=\frac{(\frac{3a}{2}+2a)\frac{h}{2}}{2}=\frac{(\frac{3a + 4a}{2})\frac{h}{2}}{2}=\frac{\frac{7a}{2}\times\frac{h}{2}}{2}=\frac{7ah}{8}\).

Step4: Find the ratio of the areas

The ratio of the area of the upper part to the area of the lower part is \(\frac{A_1}{A_2}=\frac{\frac{5ah}{8}}{\frac{7ah}{8}}=\frac{5}{7}\).

For the second question:

1. Recall the properties of a square:

• A square is a parallelogram with four equal sides and four right - angles, and its diagonals are equal and perpendicular bisectors of each other.
• In a parallelogram \(ABCD\):
• If \(AB = AD\), the parallelogram is a rhombus. If \(AC = BD\), the parallelogram is a rectangle. A parallelogram that is both a rhombus and a rectangle is a square.
• Option A: If \(AB = AD\), the parallelogram is a rhombus, and if \(AC = BD\), the parallelogram is a rectangle. A parallelogram that is a rhombus and a rectangle is a square.
• Option B: \(AB = AD\) makes it a rhombus, and \(AC\perp BD\) is a property of a rhombus, but it doesn't guarantee right - angles.
• Option C: \(m\angle A=m\angle B\) in a parallelogram means \(\angle A=\angle B = 90^{\circ}\) (since \(\angle A+\angle B = 180^{\circ}\) in a parallelogram), and \(AC = BD\) makes it a rectangle, but it doesn't guarantee all sides are equal.
• Option D: \(AC\) and \(BD\) perpendicular bisecting each other makes it a rhombus, but it doesn't guarantee right - angles.

Answer:

1. \(\frac{5}{7}\)
2. A. \(AB = AD\) and \(AC = BD\)