Question

У прямокутному трикутнику точка дотику вписаного кола ділить гіпотезу на відрізки 9 см та 6 см. Знайти площу трикутника.

Answer 1

Есть такое свойство ( доказывается в 6 секунд) - площадь прямоугольного треугольника, у которого точка касания вписанной окружности делит гипотенузу на отрезки    а и b, равна произведению а на b, т.е.
S=a*b
S=9*6=54

Answer 2

Problem Analysis

We are given a right triangle with a tangent point of an inscribed circle dividing the hypotenuse into segments of 9 cm and 6 cm. We need to find the area of the triangle.

Solution

To find the area of the triangle, we can use the formula: Area = (base * height) / 2.

In a right triangle, the hypotenuse is considered the base, and the height is the perpendicular distance from the right angle to the hypotenuse.

Let's denote the length of the hypotenuse as c, the length of the first segment as a, and the length of the second segment as b.

From the given information, we have: - Length of the first segment: a = 9 cm- Length of the second segment: b = 6 cm To find the length of the hypotenuse, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Let's denote the length of the hypotenuse as c.

Using the Pythagorean theorem, we have: a^2 + b^2 = c^2

Substituting the given values, we get: 9^2 + 6^2 = c^2 81 + 36 = c^2 117 = c^2 c = sqrt(117) ≈ 10.82 cm Now that we have the lengths of all three sides of the triangle, we can calculate the area using the formula mentioned earlier.

Using the formula Area = (base * height) / 2, we have: Area = (c * a) / 2 Area = (10.82 * 9) / 2 Area ≈ 48.69 cm^2

Therefore, the area of the triangle is approximately 48.69 cm^2.

Conclusion

The area of the right triangle with a tangent point of an inscribed circle dividing the hypotenuse into segments of 9 cm and 6 cm is approximately 48.69 cm^2.

Please let me know if you need any further assistance!