Question

Довжини сторін трикутника, виражені в сантиметрах, дорівнюють трьом послідовним натуральним числам. Знайдіть сторони цього трикутника, якщо одна з його медіан перпендикулярна до однієї з його бісектрис пожжалуйста обьясните как это делать

Answer 1

Ответ:

Розв'язання:

Нехай дано ∆АВС, СЕ - медіана, BD - бісектриса,

СЕ ┴ BD, СЕ перетинає BD в т. О.

Розглянемо ∆ВСЕ.

ВО - висота (ВО ┴ ЕС), ВО - бісектриса.

Тоді ∆ВСЕ - рівнобедрений з основою ЕС, отже, ВС = BE.

Оскільки СЕ - медіана, то BE = ЕА. 2ВС = АВ.

За умовою довжини сторін трикутника дорівнюють трьом

послідовних натуральним числам,

тоді АВ - ВС = 1 см; 2ВС - ВС = 1 см; ВС = 1 см - неможливо;

або АВ - ВС = 2 см; 2ВС - ВС = 2 см; ВС = 2 см. АВ = 4 см, АС = 3 см.

Biдповідь: ВС = 2 см, АВ = 4 см, АС = 3 см.

Пошаговое объяснение:

Answer 2

Solving for the Triangle's Sides

To solve for the sides of the triangle, we can use the information provided about the medians and bisectors. Let's break down the steps to solve this problem.

Step 1: Understanding the Problem

The problem states that the lengths of the sides of a triangle, expressed in centimeters, are three consecutive natural numbers. Additionally, one of its medians is perpendicular to one of its bisectors.

Step 2: Using Properties of Medians and Bisectors

To solve for the sides of the triangle, we can use the properties of medians and bisectors. The median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. The bisector of a triangle is a line segment that divides one of the internal angles of the triangle into two equal angles.

Step 3: Applying the Perpendicular Property

Given that one of the medians is perpendicular to one of the bisectors, we can use this information to find the sides of the triangle.

Step 4: Solving for the Sides

We can start by assuming the three consecutive natural numbers as the lengths of the sides of the triangle. Let's denote these numbers as x, x+1, and x+2.

Step 5: Using the Perpendicular Property

The median being perpendicular to the bisector means that the triangle is a right-angled triangle. We can use this information to solve for the sides.

Step 6: Applying the Pythagorean Theorem

In a right-angled 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. We can use the Pythagorean theorem to solve for the sides of the triangle.

Step 7: Solving the Equation

Using the Pythagorean theorem, we can set up the equation and solve for the value of x to find the lengths of the sides of the triangle.

I hope this helps! Let me know if you'd like to discuss this further or if there's anything else I can assist you with.