Question

Две стороны треугольника равны 12 см и 18 см а биссектриса угла между ними делит третью стороны на отрезки разность которых равна 4 см найти периметр

Answer 1

третья сторона по условию поделена на два отрезка и их разность равна 4 см,значит меньший отрезак обозначаем за х а второй за х+4, периметр треугольника равен а+в+с,значит  12+18+х+х+4=34+2х ;  2х=-34; х=-34:-2;х=17-это отрезок треугольника   вотрым действием мы находим всю сторону,17+4=21-2 отрезок,значит вся сторона равна 21+17=38,находим перимитр=38+12+18=72

Answer 2

Problem Analysis

We are given a triangle with two sides measuring 12 cm and 18 cm, and the angle bisector of the angle between these sides divides the third side into two segments whose difference is 4 cm. We need to find the perimeter of the triangle.

Solution

To solve this problem, we can use the angle bisector theorem, which states that in a triangle, the angle bisector of an angle divides the opposite side into segments that are proportional to the lengths of the adjacent sides.

Let's denote the lengths of the two segments of the third side as x and y, where x > y. According to the problem, the difference between these two segments is 4 cm. Therefore, we have the equation:

x - y = 4 ---(1)

We also know that the lengths of the two sides of the triangle are 12 cm and 18 cm. Let's denote the length of the third side as z. According to the angle bisector theorem, we have the following proportion:

x/y = 12/18 ---(2)

Now, we can solve these two equations simultaneously to find the values of x and y. Once we have x and y, we can calculate the perimeter of the triangle by adding the lengths of all three sides.

Calculation

Let's solve the equations (1) and (2) to find the values of x and y.

From equation (2), we have:

x/y = 12/18

Cross-multiplying, we get:

18x = 12y

Dividing both sides by 6, we get:

3x = 2y ---(3)

Now, let's substitute equation (3) into equation (1):

3x - 2y = 4

Simplifying, we get:

3x = 2y + 4 ---(4)

Now, we have a system of equations (3) and (4) that we can solve simultaneously.

To solve this system of equations, we can multiply equation (3) by 2 and equation (4) by 3 to eliminate the y term:

6x = 4y ---(5) 9x = 6y + 12 ---(6)

Subtracting equation (5) from equation (6), we get:

3x = 2y + 12

Simplifying, we get:

3x - 2y = 12 ---(7)

Now, we have a new equation (7) that we can solve for x and y.

Adding equation (7) to equation (4), we get:

6x = 2y + 4 + 12

Simplifying, we get:

6x = 2y + 16

Dividing both sides by 2, we get:

3x = y + 8 ---(8)

Now, we have a new equation (8) that we can solve for x and y.

Substituting equation (8) into equation (7), we get:

3x - 2(3x - 8) = 12

Simplifying, we get:

3x - 6x + 16 = 12

Combining like terms, we get:

-3x + 16 = 12

Subtracting 16 from both sides, we get:

-3x = -4

Dividing both sides by -3, we get:

x = 4/3

Substituting the value of x into equation (8), we get:

3(4/3) = y + 8

Simplifying, we get:

4 = y + 8

Subtracting 8 from both sides, we get:

-4 = y

Therefore, the values of x and y are x = 4/3 and y = -4.

However, since y represents a length, it cannot be negative. Therefore, we discard the solution y = -4.

Now, we have the values of x = 4/3 and y = 0.

Perimeter Calculation

To calculate the perimeter of the triangle, we need to find the length of the third side, z.

From equation (1), we have:

x - y = 4

Substituting the values of x and y, we get:

4/3 - 0 = 4/3

Therefore, the length of the third side, z, is 4/3 cm.

Now, we can calculate the perimeter of the triangle by adding the lengths of all three sides:

Perimeter = 12 cm + 18 cm + 4/3 cm

Simplifying, we get:

Perimeter = 38 2/3 cm

Therefore, the perimeter of the triangle is 38 2/3 cm.

Answer

The perimeter of the triangle is 38 2/3 cm.

Note: The solution provided above is based on the given information and calculations. Please double-check the calculations to ensure accuracy.