Question

20 балов решите Дві сторони трикутника дорівнюють 6 см і 8 см . Бісектриса трикутника ,що проведена до третьої сторони , поділяє її на відрізки, більший з яких дорівнює 4 см . Знайдіть периметр трикутникаСРОЧНО ПОМОГИТЕ .!!ЗАРАНИЕ СПАСИБО) не игнорьте умоляю!!!

Answer 1

Дві сторони трикутника дорівнюють 6 см і 8 см . Бісектриса трикутника ,що проведена до третьої сторони , поділяє її на відрізки, більший з яких дорівнює 4 см . Знайдіть периметр трикутника

Answer 2

Problem Analysis

We are given a triangle with two sides measuring 6 cm and 8 cm. The bisector of the triangle, which is drawn to the third side, divides it into two segments, with the larger segment measuring 4 cm. We need to find the perimeter of the triangle.

Solution

To find the perimeter of the triangle, we need to know the lengths of all three sides. We are given the lengths of two sides, but we need to find the length of the third side.

Let's assume the lengths of the two known sides are a = 6 cm and b = 8 cm. We also know that the larger segment of the third side, divided by the bisector, measures 4 cm.

Let's denote the length of the third side as c. According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Therefore, we can write the following inequalities:

a + c > b (1) b + c > a (2)

We also know that the bisector divides the third side into two segments, with the larger segment measuring 4 cm. Let's denote the lengths of these segments as x and y. Therefore, we have:

x + y = c (3) x > y (4) x - y = 4 (5)

From equation (5), we can solve for x in terms of y:

x = y + 4 (6)

Substituting equation (6) into equation (3), we get:

y + 4 + y = c 2y + 4 = c (7)

Now, we can substitute equations (4) and (7) into equations (1) and (2) to solve for c:

a + 2y + 4 > b (8) b + 2y + 4 > a (9)

Substituting the given values of a = 6 cm and b = 8 cm into equations (8) and (9), we get:

6 + 2y + 4 > 8 (10) 8 + 2y + 4 > 6 (11)

Simplifying equations (10) and (11), we have:

2y > -2 (12) 2y > -6 (13)

Dividing equations (12) and (13) by 2, we get:

y > -1 (14) y > -3 (15)

Since the length of a segment cannot be negative, we can ignore equations (14) and (15). Therefore, we can conclude that y must be greater than 0.

Now, let's solve equation (5) for y:

x - y = 4 y = x - 4 (16)

Substituting equation (16) into equation (3), we get:

x + (x - 4) = c 2x - 4 = c (17)

Now, we can substitute equation (17) into equations (1) and (2) to solve for x:

a + (2x - 4) > b (18) b + (2x - 4) > a (19)

Substituting the given values of a = 6 cm and b = 8 cm into equations (18) and (19), we get:

6 + 2x - 4 > 8 (20) 8 + 2x - 4 > 6 (21)

Simplifying equations (20) and (21), we have:

2x > 6 (22) 2x > 2 (23)

Dividing equations (22) and (23) by 2, we get:

x > 3 (24) x > 1 (25)

Since the length of a segment cannot be negative, we can ignore equations (24) and (25). Therefore, we can conclude that x must be greater than 0.

Now, let's find the values of x and y that satisfy equations (16) and (17) and are greater than 0. We can start by assuming x = 5 cm:

y = x - 4 = 5 - 4 = 1 cm (26) c = 2x - 4 = 2(5) - 4 = 6 cm (27)

Therefore, the lengths of the three sides of the triangle are: a = 6 cm b = 8 cm c = 6 cm

To find the perimeter of the triangle, we add the lengths of all three sides:

Perimeter = a + b + c = 6 + 8 + 6 = 20 cm

Therefore, the perimeter of the triangle is 20 cm.