In an isosceles triangle, the side is 10 base 5 (√6 + √2) and the angle opposite the base is 150

In an isosceles triangle, the side is 10 base 5 (√6 + √2) and the angle opposite the base is 150 find the area of the triangle.

It is known:

The ABC triangle is isosceles:
Lateral side AB = 10;
AC base = 5 * (√6 + √2);
Angle B = 150.
Find the area of ​​an isosceles triangle.

Decision.

1) The area of ​​an isosceles triangle is equal to half the product of the height and the base.

We get the formula S = 1/2 * AC * BH, BH – height.

2) Consider a triangle ABN, angle H = 90 °.

AH = 1/2 * AC = 5 * (√6 + √2) / 2;

BH = √ (AB ^ 2 – AH ^ 2) = √ (10 ^ 2 – (5 * (√6 + √2) / 2) ^ 2) = √ (100 – 25/4 * (√6 + √4 )) ^ 2 = √ (100 – 25/4 * (6 + 2√24 + 4)) = √ (100 – 25/4 * (10 + 2√24)) = √ (100 – 25/4 * 10 – 25/4 * 2√24) = √ (100 – 62.5 – 25√6) = √ (37.5 – 25√6);

3) S = 1/2 * (√6 + √2) * √ (37.5 – 25√6) = 1/2 * 3.85 * 5 = 2.5 * 3.85 = 9.625.