Question

В прямоугольном треугольнике ABC угол С =90 градусов. М-середина АС, N середина ВС, МN= 6см, угол MNC=30 градусов. Прошу, помогите. Распишите каждый шаг.

Answer 1

Расписала как смогла. Проверь!

Answer 2

Given Information:

We are given a right-angled triangle ABC, where angle C is 90 degrees. M is the midpoint of AC, N is the midpoint of BC, MN = 6 cm, and angle MNC = 30 degrees.

Step 1: Drawing the Triangle

Let's start by drawing the triangle ABC with the given information.

     B
     |\
     | \
  MN |  \ AC
     |   \
     |____\
     A  M  C

Step 2: Finding the Length of AC

Since M is the midpoint of AC, we can conclude that AM = MC. Let's denote the length of AM (or MC) as x. Therefore, AC = 2x.

Step 3: Finding the Length of BC

Since N is the midpoint of BC, we can conclude that BN = NC. Let's denote the length of BN (or NC) as y. Therefore, BC = 2y.

Step 4: Using the Pythagorean Theorem

In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. We can use this theorem to find the relationship between AC, BC, and AB.

In triangle ABC, AB is the hypotenuse. Therefore, we have:

AB^2 = AC^2 + BC^2

Substituting the values we found in Step 2 and Step 3:

AB^2 = (2x)^2 + (2y)^2

Simplifying:

AB^2 = 4x^2 + 4y^2

Step 5: Using the Given Information

We are given that MN = 6 cm and angle MNC = 30 degrees. Since MN is a line segment connecting the midpoints of two sides of a triangle, it is parallel to the third side and half its length.

Therefore, BN = NC = y, and MC = AM = x + 6.

Step 6: Applying Trigonometry

In triangle MNC, we can use trigonometry to find the relationship between the sides and angles.

We are given that angle MNC = 30 degrees. Using the sine function:

sin(30) = opposite/hypotenuse sin(30) = y/6

Simplifying:

y = 6 * sin(30) y = 3 cm

Step 7: Finding the Length of AC and BC

Using the information from Step 5, we can find the length of AC and BC.

AC = 2x = 2(x + 6) BC = 2y = 2 * 3 = 6 cm

Step 8: Finding the Length of AB

Using the Pythagorean theorem from Step 4, we can find the length of AB.

AB^2 = 4x^2 + 4y^2 AB^2 = 4(x + 6)^2 + 4(3)^2 AB^2 = 4(x^2 + 12x + 36) + 4(9) AB^2 = 4x^2 + 48x + 144 + 36 AB^2 = 4x^2 + 48x + 180

Step 9: Simplifying the Equation for AB

We can simplify the equation for AB by dividing all terms by 4.

AB^2/4 = (4x^2 + 48x + 180)/4 AB^2/4 = x^2 + 12x + 45

Step 10: Factoring the Equation for AB

We can factor the equation for AB to find the values of x.

AB^2/4 = (x + 5)(x + 9)

Since AB cannot be negative, we can discard the negative solution.

AB/2 = x + 5

Step 11: Finding the Length of AB

We know that MN is half the length of AB. Therefore, MN = AB/2.

Substituting the value of MN (6 cm):

6 = AB/2 AB = 12 cm

Final Answer:

In the given right-angled triangle ABC, with angle C = 90 degrees, the length of AB is 12 cm.