In a right-angled triangle, one of the legs is 10, and the acute angle adjacent to it is 60 degrees.

In a right-angled triangle, one of the legs is 10, and the acute angle adjacent to it is 60 degrees. Find the area of a triangle divided by √3.

Given: AB = 10 cm; ∠ABS = 60 °.
Find: S △ ABC / √3.
Since △ ABC is rectangular, then ∠BAC = 90 °. Based on the theorem on the sum of the angles of a triangle (the sum of the angles of a triangle is 180 °), ∠BCA = 180 ° – (90 ° + 60 °) = 30 °.
If in △ ABC ∠BAC = 90 °, ∠BCA = 30 °, then AB = 1/2 BC => BC = 2 * AB (property of a right-angled triangle with an angle of 30 °).
Now we can find the hypotenuse:
BC = 2 * 10 = 20 (cm).
Behind the Pythagorean theorem:
BC ^ 2 = AB ^ 2 + AC ^ 2 => AC ^ 2 = BC ^ 2 – AB ^ 2.
Find the AC leg:
AC ^ 2 = 20 ^ 2 – 10 ^ 2 = 400 – 100 = 300 (cm).
AC = Sqrt300 (cm).
Find the area of ​​the triangle:
S △ ABC = (AB * AC) / 2 = (10 * √300) / 2.
Reducing 10 and 2, we have:
S △ ABC = 5 * √300 (cm ^ 2).
Now we divide the area by the root of 3, we have:
S △ ABC / √3 = 5 * √300 / √3 = 5 * 10 = 50 (cm ^ 2).
Answer: S △ ABC / √3 = 50 cm ^ 2.