Question

В треугольнике ABC провели медиану AM. Найдите угол AMC, если углы BAC и BCA равны 45 градусов и 30 градусов соответственно. Помогите пожалуйста, срочно надо. Заранее спасибо

Answer 1

Пусть АВ=с, ВС=а
Проведем из В высоту  Н.
Н=а/2 (против угла 30 градусов)
H= c*sqrt(2)/2 (катет в равнобедренном прямоугольном треугольнике)
a=c*sqrt(2)
Но тогда в треугольниках АВМ и АВС угол В общий, а с/(а/2)=а/с
т.к. с*с=а*а/2.
Значит треугольники подобны.
Угол  АМВ=45 гр., а АМС=135 гр.

Answer 2

Finding the Angle AMC in Triangle ABC

To find the angle AMC in triangle ABC, we can use the properties of medians in a triangle.

A median of a triangle is a line segment that connects a vertex of the triangle to the midpoint of the opposite side. In this case, we have triangle ABC with the median AM.

Given that angles BAC and BCA are 45 degrees and 30 degrees respectively, we can use the properties of right-angled triangles to find the angle AMC.

Let's proceed with the solution:

1. Draw triangle ABC with the given information. Angle BAC is 45 degrees, angle BCA is 30 degrees, and median AM is drawn.

2. Since angle BAC is 45 degrees, we can determine that angle BCA is also 45 degrees. This is because the sum of the angles in a triangle is always 180 degrees.

3. Now, let's consider triangle AMC. We know that angle BCA is 45 degrees and angle BAC is 45 degrees. Therefore, angle AMC can be found by subtracting the sum of angles BCA and BAC from 180 degrees.

Angle AMC = 180 degrees - (angle BCA + angle BAC)

4. Substituting the given values, we have:

Angle AMC = 180 degrees - (45 degrees + 45 degrees)

Angle AMC = 180 degrees - 90 degrees

Angle AMC = 90 degrees

Therefore, the angle AMC in triangle ABC is 90 degrees.

I hope this helps! Let me know if you have any further questions.