Question

В равнобедренном треугольнике ABC ( AB = BC ) биссектриса AE пересекает высоту BD в точке О, причем OB/OD=3/1 . Найдите BK/KD , гду K - точка пересечения высоты AF с высотой BD. Помогите пожалуйста)

Answer 1

Задачка не очень сложная, но мне понравилась. Познавательная такая.

из Δ ВОМ  по т.Пифагора найдем ВМ
ВМ=2х√2
из подобных треугольников ВОМ и ВСД имеем
х/(2х√2) =а/4х    откуда а=х√2
из подобных треугольников АКД и ВСД имеем
КД/АД=СД/ВД
КД/х√2=х√2/4х
КД=х/2
ВК=4х-х/2=7х/2
ВК/КД=7х/2:х/2=7

Answer 2

Посмотрите рисунок с решением

Answer 3

Problem Analysis

We are given an isosceles triangle ABC, where AB = BC. The bisector AE intersects the altitude BD at point O, such that OB/OD = 3/1. We need to find the ratio BK/KD, where K is the intersection point of the altitude AF and the altitude BD.

Solution

To solve this problem, we can use the properties of similar triangles and the given ratio OB/OD = 3/1.

Let's start by labeling the points on the triangle as shown in the figure below:

         A
        / \
       /   \
      /     \
     /       \
    /         \
   /           \
  B-------------C
  |             |
  |             |
  |             |
  D-------------E

We know that AB = BC, so triangle ABC is an isosceles triangle. Let's denote the length of AB (or BC) as c and the length of AC as b.

Now, let's consider triangle ABD. Since AE is the bisector of angle BAC, it divides the base BD into two equal segments, i.e., BD = AD.

Using the given ratio OB/OD = 3/1, we can write OB = 3x and OD = x, where x is a positive constant.

Let's denote the length of BK as y and the length of KD as z.

Now, we can apply the properties of similar triangles to find the ratio BK/KD.

In triangle ABD, we have: - BD = AD (isosceles triangle) - OB/OD = 3/1

In triangle ABK, we have: - BK/BD = AK/AD (similar triangles)

In triangle AKD, we have: - KD/BD = AK/AD (similar triangles)

From the above equations, we can write: - BK/BD = AK/AD - KD/BD = AK/AD

Since BD = AD, we can simplify the above equations to: - BK/BD = AK/BD - KD/BD = AK/BD

Canceling out the common term BD, we get: - BK = AK - KD = AK

Therefore, the ratio BK/KD is 1:1.

Answer

The ratio BK/KD is 1:1.

Explanation

In the given isosceles triangle ABC, where AB = BC, the ratio BK/KD is 1:1. This can be proved using the properties of similar triangles and the given ratio OB/OD = 3/1. By applying the properties of similar triangles, we find that BK = AK and KD = AK, which implies that the ratio BK/KD is 1:1.