Question

Прямая, проведенная через вершину А тр. АВС , перпендикулярна его медиане СМ и делит ее пополам . найдиье сторону АС, если АВ =18см( если что рисунка нет )

Answer 1

Решение 
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Answer 2

Problem Analysis

We are given a triangle ABC, and it is stated that a line passing through vertex A is perpendicular to its median CM and divides it in half. We need to find the length of side AC, given that AB = 18 cm.

Solution

To solve this problem, let's analyze the given information. We know that the line passing through vertex A is perpendicular to the median CM and divides it in half. This means that the line passing through A is also the altitude of the triangle ABC.

We can use the properties of right-angled triangles to find the length of side AC. Let's denote the midpoint of side BC as M. Since the line passing through A is perpendicular to CM and divides it in half, we can conclude that triangle AMC is a right-angled triangle with AC as the hypotenuse.

Using the Pythagorean theorem, we can find the length of AC. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Let's denote the length of AC as x. We know that AB = 18 cm. Using the Pythagorean theorem, we can write the following equation:

x^2 = (CM/2)^2 + AB^2

Now, let's find the length of CM. The median of a triangle divides the opposite side into two equal parts. Since the line passing through A divides CM in half, we can conclude that CM = 2 * AM.

To find the length of AM, we can use the fact that the median divides the triangle into two triangles with equal areas. The area of triangle ABC can be calculated using the formula:

Area = (1/2) * base * height

Since the median CM divides the triangle into two triangles with equal areas, the area of triangle AMC is equal to the area of triangle BMC. The base of triangle AMC is AC, and the height is AM. The base of triangle BMC is BC, and the height is BM.

Using the formula for the area of a triangle, we can write the following equation:

(1/2) * AC * AM = (1/2) * BC * BM

Since BM is half the length of BC, we can write BM = BC/2. Substituting this into the equation above, we get:

(1/2) * AC * AM = (1/2) * BC * (BC/2)

Simplifying this equation, we get:

AC * AM = (BC^2)/4

Now, let's substitute the value of CM = 2 * AM into the equation above:

AC * 2 * AM = (BC^2)/4

Simplifying further, we get:

AC * CM = (BC^2)/4

Since we know that BC = 2 * CM, we can substitute this into the equation above:

AC * CM = ((2 * CM)^2)/4

Simplifying this equation, we get:

AC * CM = CM^2

Now, let's substitute the value of CM = 2 * AM into the equation above:

AC * 2 * AM = (2 * AM)^2

Simplifying further, we get:

AC * 2 * AM = 4 * AM^2

Dividing both sides of the equation by 2 * AM, we get:

AC = 2 * AM

Now, let's substitute the value of AM = CM/2 into the equation above:

AC = 2 * (CM/2)

Simplifying this equation, we get:

AC = CM

Therefore, we can conclude that the length of side AC is equal to the length of the median CM.

Now, let's find the length of AC using the given information. We are given that AB = 18 cm. Let's find the length of CM using the formula CM = 2 * AM.

To find the length of AM, we can use the fact that the median divides the triangle into two triangles with equal areas. The area of triangle ABC can be calculated using the formula:

Area = (1/2) * base * height

Substituting the values, we get:

Area = (1/2) * AB * CM

Since the area of triangle AMC is equal to the area of triangle BMC, we can write:

(1/2) * AB * CM = (1/2) * BC * BM

Since BM is half the length of BC, we can write BM = BC/2. Substituting this into the equation above, we get:

(1/2) * AB * CM = (1/2) * BC * (BC/2)

Simplifying this equation, we get:

AB * CM = (BC^2)/4

Substituting the value of BC = 2 * CM, we get:

AB * CM = ((2 * CM)^2)/4

Simplifying further, we get:

AB * CM = CM^2

Now, let's substitute the value of AB = 18 cm into the equation above:

18 * CM = CM^2

Rearranging the equation, we get:

CM^2 - 18 * CM = 0

Factoring out CM, we get:

CM * (CM - 18) = 0

Since CM cannot be zero, we can conclude that CM - 18 = 0. Solving for CM, we get:

CM = 18

Therefore, the length of CM is 18 cm, which is also the length of AC.

Answer

The length of side AC is 18 cm.

Explanation

We can solve this problem by using the properties of right-angled triangles and the fact that the line passing through vertex A is perpendicular to the median CM and divides it in half. By applying the Pythagorean theorem and the properties of triangles, we can find that the length of side AC is equal to the length of the median CM, which is 18 cm.