Question

Боковая сторона АВ равнобедренного треугольника АВС равна 5 см,основание АС равно 6 см,центр вписанной окружности лежит на высоте ВН и удален от вершины В на 2,5см. Найти радиус вписанной окружности.

Answer 1

Дано:
АВ=ВС=5 см
АС=6 см
ВО=2,5 см
ОН-?
Решение:
Рассмотрим треугольник ВАН
угол АНВ=90* => треугольник прямоугольный
ВН=корень 5^2-3^2 = корень 25-9 = корень 16=4
ОН= 4-2,5 =1,5 (см)

Answer 2

Calculation of the Radius of the Inscribed Circle

To find the radius of the inscribed circle in a triangle, we can use the formula:

radius = (area of the triangle) / (semiperimeter of the triangle)

Let's calculate the area of the triangle and the semiperimeter first.

Given: - Side AB = 5 cm (the lateral side of the isosceles triangle) - Base AC = 6 cm - The center of the inscribed circle lies on the height BN and is 2.5 cm away from vertex B.

Calculation Steps:

1. Calculate the height of the triangle using the Pythagorean theorem: - The height BN is perpendicular to the base AC, so we can use the Pythagorean theorem to find its length. - The length of the height BN is the difference between the length of the lateral side AB and the distance from the center of the inscribed circle to vertex B. - The height BN can be calculated as follows: ``` BN = AB - distance from center to vertex B = 5 cm - 2.5 cm = 2.5 cm ```

2. Calculate the area of the triangle using the formula: ``` area = (base * height) / 2 = (6 cm * 2.5 cm) / 2 = 15 cm² ```

3. Calculate the semiperimeter of the triangle using the formula: ``` semiperimeter = (AB + BC + AC) / 2 = (5 cm + 5 cm + 6 cm) / 2 = 8 cm ```

4. Calculate the radius of the inscribed circle using the formula: ``` radius = area / semiperimeter = 15 cm² / 8 cm = 1.875 cm ```

Therefore, the radius of the inscribed circle in the given triangle is 1.875 cm.