Question

ДАМ 70 БАЛОВ BK и AR — медианы. BR= 8 м; AK= 14 м; RK= 20 м. Найти: P(ABC). Каковы длины сторон (ответы укажите через запятую в той последовательности, в которой они указаны) ? AC= м; BC= м; AB= м. P(ABC)= м. Напишите ответ

Answer 1

Пошаговое объяснение:

АС=28 м

ВС=16 м

АВ=20 м

Р(АВС) =64 м

Answer 2

Problem Analysis

We are given the following information: - BK and AR are medians of triangle ABC. - BR = 8 m, AK = 14 m, and RK = 20 m.

We need to find: - The length of sides AC, BC, and AB. - The perimeter of triangle ABC.

Solution

To find the lengths of sides AC, BC, and AB, we can use the fact that BK and AR are medians. In a triangle, medians divide each other in a 2:1 ratio. This means that BK divides AC into two segments, with the segment towards A being twice as long as the segment towards C. Similarly, AR divides BC into two segments, with the segment towards B being twice as long as the segment towards C.

Let's denote the length of segment AC towards A as x. Then, the length of segment AC towards C is 2x. Similarly, let's denote the length of segment BC towards B as y. Then, the length of segment BC towards C is 2y.

Using this information, we can set up the following equations: - x + 2x + 14 = AC (equation 1) - y + 2y + 8 = BC (equation 2)

Simplifying equation 1, we get: - 3x + 14 = AC

Simplifying equation 2, we get: - 3y + 8 = BC

To find the length of AB, we can use the fact that AB is the third side of triangle ABC. We can use the Pythagorean theorem to find the length of AB.

Let's denote the length of AB as z. Then, we can set up the following equation: - z^2 = (AC)^2 + (BC)^2

Now, let's solve these equations to find the lengths of sides AC, BC, and AB.

Calculation

Using equation 1, we can solve for x: - 3x + 14 = AC - 3x = AC - 14 - x = (AC - 14) / 3

Using equation 2, we can solve for y: - 3y + 8 = BC - 3y = BC - 8 - y = (BC - 8) / 3

Substituting the values of x and y into the equation for AB, we get: - z^2 = ((AC - 14) / 3)^2 + ((BC - 8) / 3)^2

Now, let's substitute the given values of BR, AK, and RK into the equations to find the lengths of sides AC, BC, and AB.

Substituting BR = 8 into equation 2, we get: - 3y + 8 = BC - 3y + 8 = 8 - 3y = 0 - y = 0

Substituting AK = 14 into equation 1, we get: - 3x + 14 = AC - 3x + 14 = 14 - 3x = 0 - x = 0

Substituting RK = 20 into equation 2, we get: - 3y + 8 = BC - 3y + 8 = 20 - 3y = 12 - y = 4

Substituting AK = 14 into equation 1, we get: - 3x + 14 = AC - 3x + 14 = 14 - 3x = 0 - x = 0

Now, let's substitute the values of x and y into the equation for AB to find the length of AB: - z^2 = ((AC - 14) / 3)^2 + ((BC - 8) / 3)^2 - z^2 = ((0 - 14) / 3)^2 + ((4 - 8) / 3)^2 - z^2 = (-14 / 3)^2 + (-4 / 3)^2 - z^2 = (196 / 9) + (16 / 9) - z^2 = 212 / 9 - z = sqrt(212 / 9) - z ≈ 4.62

Therefore, the length of side AB is approximately 4.62 m.

To find the perimeter of triangle ABC, we can add the lengths of sides AC, BC, and AB: - P(ABC) = AC + BC + AB - P(ABC) = 0 + 4 + 4.62 - P(ABC) ≈ 8.62

Therefore, the perimeter of triangle ABC is approximately 8.62 m.

Answer

The lengths of the sides of triangle ABC, in the order they were mentioned, are: - AC = 0 m - BC = 4 m - AB = 4.62 m

The perimeter of triangle ABC is approximately 8.62 m.