Question

Помогите пожалуйста, решить срочно. Через середину диагонали BD квадрата ABCD проведена прямая пересекающая стороны AB и CD в точках P и K. Требуется доказать что BKDP-параллелограмм. Найти S четырехугольника если AP=2 см, а KD=6 см

Answer 1

AB||CD (стороны квадрата)
∠BOP=∠DOK (вертикальные углы)
∠PBD=∠BDC (накрест лежащие углы при AB||CD)
BO=OD (по условию)
△BOP=△DOK (по стороне и двум прилежащим к ней углам)
PB=KD, PB||KD => BKDP - параллелограмм (две противоположные стороны одновременно равны и параллельны).

BC⊥CD (стороны квадрата)
BC - высота BKDP.

AB=CD=BC (стороны квадрата)
AB=CD <=> AP+PB=CK+KD <=> AP=CK
BC=CD =CK+KD =AP+KD =2+6 =8 (см)
S(BKDP)= KD*BC =6*8 =48 (см^2)

Answer 2

Problem Analysis

To prove that quadrilateral BKDP is a parallelogram, we need to show that opposite sides are parallel. Additionally, we are given the lengths of AP and KD and need to find the area of quadrilateral BKDP.

Solution

Let's start by analyzing the given information and proving that BKDP is a parallelogram.

1. We are given that a line passing through the midpoint of diagonal BD intersects sides AB and CD at points P and K, respectively. 2. Since the line passes through the midpoint of BD, it divides BD into two equal segments: BP and PD. 3. By the midpoint theorem, we know that the line passing through the midpoint of a side of a triangle is parallel to the opposite side. Therefore, BP is parallel to AD, and PD is parallel to BC. 4. Since opposite sides of quadrilateral BKDP are parallel, we can conclude that BKDP is a parallelogram.

Now, let's find the area of quadrilateral BKDP.

1. We are given that AP = 2 cm and KD = 6 cm. 2. Since BP is parallel to AD, we can use the proportionality of corresponding sides in similar triangles to find the length of BP. - Triangle ABP is similar to triangle ADK. - Therefore, we can set up the following proportion: AP/AD = BP/DK. - Substituting the given values, we have: 2/AD = BP/6. - Solving for BP, we get: BP = (2/AD) * 6. 3. Similarly, since PD is parallel to BC, we can use the proportionality of corresponding sides in similar triangles to find the length of PD. - Triangle PDK is similar to triangle BCD. - Therefore, we can set up the following proportion: DK/BC = PD/CD. - Substituting the given values, we have: 6/BC = PD/6. - Solving for PD, we get: PD = (6/BC) * 6. 4. Now that we have the lengths of BP and PD, we can calculate the area of quadrilateral BKDP using the formula for the area of a parallelogram. - The area of a parallelogram is given by the formula: Area = base * height. - In this case, the base of the parallelogram is BP, and the height is PD. - Therefore, the area of quadrilateral BKDP is: Area = BP * PD.

Let's calculate the values of BP, PD, and the area of quadrilateral BKDP using the given information.

Calculation

Given: AP = 2 cm KD = 6 cm

1. Calculating BP: - BP = (2/AD) * 6

2. Calculating PD: - PD = (6/BC) * 6

3. Calculating the area of quadrilateral BKDP: - Area = BP * PD

Now, let's substitute the given values and calculate the results.

Results

Given: AP = 2 cm KD = 6 cm

1. Calculating BP: - BP = (2/AD) * 6

2. Calculating PD: - PD = (6/BC) * 6

3. Calculating the area of quadrilateral BKDP: - Area = BP * PD

Please provide the values of AD and BC so that we can calculate the final results.