Question

В трапеции АВСD с основаниями АD и ВС диагонали пересекаются в точке О, ВО:ОD =2:3, АС=25см. Найдите АО и ОС. Мне важно решение, пожалуйста расспишите​

Answer 1

Рассмотрим трапецию ABCD. У данной трапеции ∠BOC = ∠AOD данные углы равны как вертикальные. Так-как это трапеция, то основания трапеции параллельны то есть

BC ∥ AD. Тогда ∠BCA = ∠CAD как накрест лежащие углы при параллельных прямых BC, AD и секущей AC. Рассмотрим △BOC и △AOD, которые являются подобными по 1 признаку (по двум углам). Тогда можем составить отношение сторон:

$\frac{DO}{BO}=\frac{AO}{CO}=\frac{AD}{BC}$.

Так-как по условию AC = 25, то AO = AC - CO = 25 - CO.

BO:OD = 2:3, тогда BO = 2x, OD = 3x Отсюда получаем.

$\frac{3x}{2x}=\frac{25-CO}{CO}$ (Отношение AD/BC нам не понадобится, поэтому его можем убрать).

$\frac{25-CO}{CO}=\frac{3}{2};\\\\2(25-CO)=3CO;\\\\50-2CO=3CO;\\\\50=5CO;\\\\CO=10$

Отсюда: $AO=25-CO=25-10=15$.

Ответ: 15; 10

Answer 2

Problem Analysis

We are given a trapezoid ABCD with bases AD and BC. The diagonals intersect at point O, and we are given that the ratio of BO to OD is 2:3. We are also given that AC is 25 cm. We need to find the lengths of AO and OC.

Solution

To solve this problem, we can use the properties of similar triangles and the fact that the diagonals of a trapezoid bisect each other.

Let's denote the length of AO as x and the length of OC as y.

Since the diagonals of a trapezoid bisect each other, we can write the following equations:

1. AO/OD = BO/OC (Diagonals bisect each other) 2. BO/OD = 2/3 (Given ratio)

From equation 1, we can rewrite it as:

AO/OD = BO/OC x/OD = BO/OC

From equation 2, we know that BO/OD = 2/3. We can substitute this into equation 1:

x/OD = 2/3

Now, let's consider triangle AOC. We know that AC is 25 cm. Using the property of similar triangles, we can write the following equation:

AO/AC = OD/OC

Substituting the values we know:

x/25 = OD/OC

We can rearrange this equation to solve for OD:

OD = (x/25) * OC

Substituting this into the equation x/OD = 2/3:

x/((x/25) * OC) = 2/3

Simplifying this equation:

3x = 2(x/25) * OC 3x = (2x/25) * OC

Now, we can solve for OC:

3x = (2x/25) * OC 3x = (2/25) * x * OC OC = (3 * 25) / 2 OC = 37.5 cm

Finally, we can substitute the value of OC back into the equation x/OD = 2/3 to solve for x:

x/OD = 2/3 x/OD = 2/3 x/((x/25) * OC) = 2/3 x/((x/25) * 37.5) = 2/3 x/(x/2) = 2/3 x = (x/2) * (2/3) x = x/3 3x = x 2x = 0 x = 0

Therefore, the length of AO is 0 cm.

To summarize: - AO = 0 cm - OC = 37.5 cm

Please note that the length of AO being 0 cm indicates that the point O lies on the line AD, and the trapezoid degenerates into a triangle.