Question

В треугрльнике ABC угол С=90, угол САВ=60. АВ=6 см. Найти биссектрису АМ Помогите, СРОЧНО!!!!

Answer 1

Problem Analysis

We are given a triangle ABC, where angle C is 90 degrees and angle CAB is 60 degrees. We are also given that AB is 6 cm. We need to find the bisector of angle A, which passes through point M.

Solution

To find the bisector of angle A, we can use the properties of a right-angled triangle and the angle bisector theorem.

Let's start by drawing the triangle ABC:

     A
     |\
     | \
  6  |  \  x
     |   \
     |    \
     |_____\
     B   90° C

We are given that angle CAB is 60 degrees and angle C is 90 degrees. To find the bisector of angle A, we need to find the length of AM.

Using the properties of a right-angled triangle, we can find the length of AC using the sine function:

sin(60°) = AC / AB

Simplifying the equation, we have:

AC = AB * sin(60°)
   = 6 * sin(60°)

Now, let's find the length of AM using the angle bisector theorem. According to the theorem, the ratio of the lengths of the segments formed by the angle bisector is equal to the ratio of the lengths of the opposite sides of the triangle.

AM / MC = AB / BC

Substituting the known values, we have:

AM / (AC + MC) = 6 / BC

Since we know the length of AC and BC, we can solve for AM:

AM / (AC + AM) = 6 / BC

Simplifying the equation, we have:

AM = (6 * AC) / (BC - 6)

Now, let's substitute the values of AC and BC:

AM = (6 * (6 * sin(60°))) / (BC - 6)

To find the value of BC, we can use the Pythagorean theorem:

BC^2 = AB^2 + AC^2
BC^2 = 6^2 + (6 * sin(60°))^2
BC^2 = 36 + 36
BC^2 = 72
BC = sqrt(72)

Substituting the value of BC, we have:

AM = (6 * (6 * sin(60°))) / (sqrt(72) - 6)

Now, let's calculate the value of AM using the given values:

AM = (6 * (6 * sin(60°))) / (sqrt(72) - 6)
   ≈ 6.928 cm

Therefore, the length of the bisector AM is approximately 6.928 cm.

Answer

The length of the bisector AM is approximately 6.928 cm.

Please note that the above calculations are based on the given information and assumptions. If there are any additional constraints or information, the solution may vary.