Question

В треугольнике АВС сторона АВ=4см, ВС=7см, АС=6см, а в треугольнике MNK сторона MK=8см, NK=14см, MN=12см. Найдите углы треугольника MNK, если угол А=80 градусов, угол В=60 градусов. Помогите пожалуйста!!!!!

Answer 1

Реугольник SAB подобен треуг. SMN, т. к. Угол ASB = углу MSN и SA:SM = SB:SN = 3:2 по условию.  
Раз треугольники подобны, то AB:MN = 3:2
Аналогично доказываем, что BC:NK = 3:2 и AC:MK = 3:2
Из всего этого получаем что треуг ABC подобен треуг/ MNK, т. к. соответственные стороны у них пропорциональны с коеф пропорциональности 3:2

А у подобных треуг площади относятся, как квадрат коеф подобия, отсюда S MNK = S ABC / ( 3/2 )^2 = 10 / ( 9/4 ) = 40 / 9 = 4 4/9. Округляй сам

Answer 2

Given Information

We are given the following information about two triangles:

Triangle ABC: - Side AB = 4 cm - Side BC = 7 cm - Side AC = 6 cm - Angle A = 80 degrees - Angle B = 60 degrees

Triangle MNK: - Side MK = 8 cm - Side NK = 14 cm - Side MN = 12 cm

We need to find the angles of triangle MNK.

Solution

To find the angles of triangle MNK, we can use the Law of Cosines and the Law of Sines.

# Law of Cosines

The Law of Cosines states that in a triangle with sides a, b, and c, and angle C opposite side c, the following equation holds:

c^2 = a^2 + b^2 - 2ab * cos(C)

Applying the Law of Cosines to triangle MNK, we can find the angle MKN:

MK^2 = MN^2 + NK^2 - 2 * MN * NK * cos(MKN)

Substituting the given values:

8^2 = 12^2 + 14^2 - 2 * 12 * 14 * cos(MKN)

Simplifying the equation:

64 = 144 + 196 - 336 * cos(MKN)

336 * cos(MKN) = 144 + 196 - 64

336 * cos(MKN) = 276

cos(MKN) = 276 / 336

cos(MKN) ≈ 0.8214

Using the inverse cosine function, we can find the value of angle MKN:

MKN ≈ arccos(0.8214)

MKN ≈ 34.98 degrees

# Law of Sines

The Law of Sines states that in a triangle with sides a, b, and c, and angles A, B, and C opposite their respective sides, the following equation holds:

a / sin(A) = b / sin(B) = c / sin(C)

Applying the Law of Sines to triangle MNK, we can find the remaining angles:

sin(MKN) / MN = sin(MNK) / NK

sin(MKN) / 12 = sin(MNK) / 14

Cross-multiplying:

14 * sin(MKN) = 12 * sin(MNK)

sin(MKN) ≈ (12 * sin(MNK)) / 14

Using the identity sin(180 - x) = sin(x), we can find the value of angle MNK:

sin(MNK) = sin(180 - MNK)

sin(MNK) ≈ sin(180 - 34.98)

sin(MNK) ≈ sin(145.02)

MNK ≈ 145.02 degrees

Finally, we can find the remaining angle MKN:

MKN + MNK + MK = 180

34.98 + 145.02 + 180 = 360

MK ≈ 180 degrees

Answer

The angles of triangle MNK are approximately: - MKN ≈ 34.98 degrees - MNK ≈ 145.02 degrees - MK ≈ 180 degrees

Please note that the values are approximate due to rounding.

Let me know if there's anything else I can help you with!