Question

ПОЖАЛУЙСТА ПОМОГИТЕ! Точка M — середина стороны CD параллелограмма ABCD; H — проекция точки B на прямую AM . Найдите CH, если AB = a, BC=b.​

Answer 1

E - середина AB

Противоположные стороны параллелограмма параллельны и равны.

AB=CD => AB/2=CD/2 => AE=CM

AECM - параллелограмм (AE||CM, AE=CM)

=> CE||AM

BH⊥AM => BH⊥CE

BE=EA => BF=FH (т Фалеса)

CF - высота и медиана в △BCH => △BCH - р/б

CH=BC=b

Answer 2

Problem Analysis

We are given a parallelogram ABCD, with M as the midpoint of side CD, and H as the projection of point B onto the line AM. We need to find the length of CH, given that AB = a and BC = b.

Solution

To find the length of CH, we can use the properties of parallelograms and right triangles.

1. Since ABCD is a parallelogram, opposite sides are equal. Therefore, AD = BC = b.

2. Let's consider triangle ABM. Since M is the midpoint of CD, we can conclude that DM = MC = b/2.

3. Since H is the projection of point B onto the line AM, triangle ABH is a right triangle. Therefore, we can use the Pythagorean theorem to find the length of AH.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

In triangle ABH, the sides are AB = a, BH = CH, and AH is the hypotenuse.

Applying the Pythagorean theorem, we have:

AH^2 = AB^2 - BH^2

4. Now, let's find the length of AH. We can use the fact that M is the midpoint of CD to find the length of AM.

Since M is the midpoint of CD, we have CM = DM = b/2. Therefore, the length of AM is:

AM = AD - DM = b - b/2 = b/2

5. Substituting the values of AB = a and AM = b/2 into the equation from step 3, we have:

AH^2 = a^2 - BH^2

6. Finally, let's find the length of CH. Since H is the projection of point B onto the line AM, we have:

CH = BH - BC

Substituting the value of BH from step 5, we have:

CH = sqrt(a^2 - AH^2) - b

Answer

The length of CH is given by the formula:

CH = sqrt(a^2 - (b/2)^2) - b

Please note that this formula assumes that the given values of a and b are positive.