Question

Сторони трикутника АВС а=21 м b=14 м с=15 м. Знайти відношення відрізків на трьох бісектрисах які утворюются точкою перетину бісектрис трикутника АВС

Answer 1

 Точкой пересечения биссектрисы делятся в отношении суммы сторон треугольника, образующих угол, в котором проведена биссектриса, к третьей стороне: $\frac{x}{y} = \frac{a+b}{c}$.
Здесь х и у - части биссектрисы, считая от вершины треугольника.
Обозначим биссектрисы: АЕ, ВК и СД, точка их пересечения О.
Тогда отношения отрезков биссектрис:
$\frac{OB}{OK} = \frac{15+21}{14} = \frac{36}{14} = \frac{18}{7}$
$\frac{OC}{OD} = \frac{21+14}{15} = \frac{35}{15} = \frac{7}{3}$
$\frac{AO}{OE} = \frac{15+14x}{21} = \frac{29}{21}$

Answer 2

Finding the Ratio of Segments on the Triangle's Bisectors

To find the ratio of segments on the three bisectors formed by the intersection point of the bisectors of triangle ABC, we can use the angle bisector theorem. According to this theorem, the ratio of the lengths of the segments formed by the intersection point on each bisector is equal to the ratio of the lengths of the sides opposite those bisectors.

Let's denote the intersection point of the bisectors as point O. The lengths of the sides of triangle ABC are given as follows: - Side AB: a = 21 m - Side BC: b = 14 m - Side AC: c = 15 m

To find the ratio of the segments on the bisectors, we need to calculate the lengths of the sides opposite those bisectors. Let's denote these lengths as x, y, and z.

Calculation

To calculate the lengths of the sides opposite the bisectors, we can use the angle bisector theorem. According to this theorem, the ratio of the lengths of the sides is equal to the ratio of the lengths of the segments formed by the intersection point on each bisector.

Let's calculate the lengths of the sides opposite the bisectors:

- Side opposite the bisector from point A: x - Side opposite the bisector from point B: y - Side opposite the bisector from point C: z

According to the angle bisector theorem, we have the following ratios:

x / 21 = y / 14 = z / 15

To find the values of x, y, and z, we can solve this system of equations. Let's solve it:

From the first equation, we can express x in terms of y:

x = (21 * y) / 14

From the second equation, we can express z in terms of y:

z = (15 * y) / 14

Substituting these values into the third equation, we get:

(21 * y) / 14 / 21 = (15 * y) / 14 / 15

Simplifying the equation, we have:

y / 14 = y / 14

This equation is true for any value of y. Therefore, we can choose any value for y, and the ratios will remain the same.

Let's choose y = 14. This means that x = 21 and z = 15.

Ratio of Segments on the Bisectors

Therefore, the ratio of the segments on the three bisectors formed by the intersection point of the bisectors of triangle ABC is:

x : y : z = 21 : 14 : 15

Please note that the values of x, y, and z are proportional to the lengths of the sides opposite the bisectors, and the ratio remains the same regardless of the chosen value for y.

I hope this helps! Let me know if you have any further questions.