Line segment AB is the diameter of a circle centered on. Point C lies on a circle, AO = AC. Calculate the area

| August 8, 2021 | education

Line segment AB is the diameter of a circle centered on. Point C lies on a circle, AO = AC. Calculate the area of triangle ABC and the distance from point C to line AB if AB = 12 cm.

Let’s take a look at the figure to solve it.

Since the angle inscribed in the circle based on the diameter is 90, the angle ACB is straight, and the triangle ABC is rectangular, in which AB = 12 cm, CB = OB = AB / 2 = 6 cm.

By the Pythagorean theorem, we find the leg AC of the right-angled triangle ABC.

AC ^ 2 = AB ^ 2 – BC ^ 2 = 12 ^ 2 – 6 ^ 2 = 144 – 36 = 108 = 36 * 3.

AC = √36 * 3 = 6 * √3.

The area of ​​a right-angled triangle is:

S = (AC * CB) / 2 = (6 * √3 * 6) / 2 = 18 * √3 cm2.

The desired distance from point C to line AB is the perpendicular to AB, therefore CE is the height of the right-angled triangle ABC.

Also, the area of ​​a right-angled triangle is equal to half the product of the base by the height:

S = (AB * CE) / 2.

18 * √3 = 12 * CE / 2.

CE = 2 * 18 * √3 / 12 = 3 * √3 cm.

Answer: The area is 18 * √3 cm2, the distance is 3 * √3 cm.



One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.