A circle is inscribed in a right-angled triangle. the point of contact divides one of the legs into segments 3 cm

| February 10, 2021 | education

A circle is inscribed in a right-angled triangle. the point of contact divides one of the legs into segments 3 cm and 5 cm long, counting from the top of the right angle. find the hypotenuse and the second leg of the triangle.

We use the properties of a tangent drawn from one point, according to which, the segments of the tangents are equal to each other.

Then ВK = BH = 3 cm, CH = CM = 5 cm, AK = AM.

Let the length of the segment AK = AM = X cm, then the length of the segment AB = (X + 3) cm, AC = (X + 5) cm.

In a right-angled triangle ABC, according to the Pythagorean theorem, AC ^ 2 = AB ^ 2 + BC ^ 2.

(X + 5) ^ 2 = (X + 3) ^ 2 + 8 ^ 2.

X ^ 2 + 10 * X + 25 = X ^ 2 + 6 * X + 9 + 64.

4 * X = 64 + 9 – 25 = 48.

X = AK = AM = 48/4 = 12 cm.

Then AB = 3 + 12 = 15 cm.

AC = 5 + 12 = 17 cm.

Answer: The length of the hypotenuse is 17 cm, the second leg is 15 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.