The middle side of the triangle is 1 more than the smallest and 1 less than the largest side.
| February 4, 2021 | education
The middle side of the triangle is 1 more than the smallest and 1 less than the largest side. The cosine of the average angle is 2/3. Find the perimeter of this triangle.
Let the length of the middle segment of the triangle be X cm, BC = X, then AB = (X – 1), AC = (X + 1).
To solve the problem, we use the cosine theorem for a triangle.
BC ^ 2 = AB ^ 2 + AC ^ 2 – 2 * AB * AC * CosA.
CosA = ((X – 1) ^ 2 + (X + 1) ^ 2 – X ^ 2) / 2 * (X – 1) * (X + 1).
2/3 = (X ^ 2 – 2 * X + 1 + X ^ 2 + 2 * X +1 – X ^ 2) / (2 * X ^ 2 – 2).
2/3 = (X ^ 2 + 2) / (2 * X ^ 2 – 2).
3 * X ^ 2 + 6 = 4 * X ^ 2 – 4.
X ^ 2 = 10.
X = √10.
BC = √10.
AB = √10 – 1.
AC = √10 + 1.
P = √10 + √10 – 1 + √10 + 1 = 3 * √10.
Answer: The perimeter of a triangle is 3 * √10.
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