Point M was placed inside the equilateral triangle, which is respectively at distances of 3 cm

Point M was placed inside the equilateral triangle, which is respectively at distances of 3 cm, 4 cm and 5 cm from its vertices. Find the side length of this triangle.

Given:

DABC

AB = BC = AC

tM lies inside DABC

AM = 3 cm

BM = 5 cm

CM = 4cm

Find: AB

Decision:

Let AB = BC = AC = x

Cosine theorem: The square of any side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of those sides by the cosine of the angle between them.

In DACM, by the cosine theorem, we find ∠ACM:

АМ ^ 2 = АС ^ 2 + МС ^ 2 – 2 × АС × МС × cos∠АСМ

3 ^ 2 = x ^ 2 + 4 ^ 2 – 2 × x × 4 × cos∠АСМ

9 = x ^ 2 + 16 – 8x × cos∠АСМ

cos∠АСМ = (х2 + 7) / 8х

In DBCM, by the cosine theorem, we find ∠ACM:

ВМ ^ 2 = ВС ^ 2 + МС ^ 2 – 2 × ВС × МС × cos∠АСМ

5 ^ 2 = x ^ 2 + 4 ^ 2 – 2 × x × 4 × cos∠АСМ

25 = x ^ 2 + 16 – 8x × cos∠АСМ

cos∠АСМ = (х2 – 9) / 8х

Since DABS is equilateral, then ∠АСВ = ∠АСМ + ∠ВСМ = 60˚ = π / 3

Let cos∠АСМ = cos φ1, cos∠АСМ = cos φ2

Now we substitute the values ​​of the cosines into this equation, getting

(x ^ 2 + 7) ^ 2 + (x ^ 2 – 9) ^ 2 – (x ^ 2 + 16) (x ^ 2 + 9) = 34 ⋅ 84×2

The biquadratic equation is obtained

x ^ 4 – 50x ^ 2 + 193 = 0

One of the values ​​is discarded, since it is easy to deduce from the triangle inequality that

2x = AB + AC> PB + PC = 9.

Therefore, x2> 81/4> 20, and therefore there can be no minus sign in front of the root.

Answer: The length of the side of the triangle DABS is equal to