Question

Периметр равнобедренного треугольника равен 32 см. Биссектриса угла, протеволежащего основания, делит треугольник на два треугольника, периметры которых равны по 24 см. Найди эту биссектрису

Answer 1

будет 12 см..........

Answer 2

Perimeter of an Isosceles Triangle

To find the length of the bisector of an angle in an isosceles triangle, we first need to understand some properties of the triangle.

Let's denote the length of the two congruent sides of the triangle as a and the length of the base as b. The perimeter of the triangle is given as 32 cm, so we can write the equation:

2a + b = 32 .

Bisector of the Angle

The bisector of the angle opposite the base divides the triangle into two smaller triangles. The perimeters of these two triangles are given as 24 cm each. Let's denote the length of the bisector as x.

Using the properties of the bisector, we can write the following equations:

a + x = 24 (perimeter of the left triangle).

a + x = 24 (perimeter of the right triangle).

Simplifying these equations, we get:

2a + 2x = 48.

Solving the Equations

Now we have two equations:

2a + b = 32 (Equation 1).

2a + 2x = 48 (Equation 2).

To find the value of x, we need to eliminate a from these equations. Subtracting Equation 1 from Equation 2, we get:

(2a + 2x) - (2a + b) = 48 - 32.

Simplifying further, we have:

2x - b = 16.

Rearranging the equation, we get:

2x = 16 + b.

Finally, dividing both sides of the equation by 2, we find:

x = (16 + b) / 2.

Finding the Value of b

To find the value of b, we can substitute the given perimeter of the triangle into Equation 1:

2a + b = 32.

Since we know that the perimeters of the two smaller triangles are equal to 24 cm each, we can write:

a + x = 24.

Substituting a = 24 - x into Equation 1, we get:

2(24 - x) + b = 32.

Simplifying, we have:

48 - 2x + b = 32.

Rearranging the equation, we find:

b = 32 - 48 + 2x.

Simplifying further, we have:

b = 2x - 16.

Substituting b into the Equation for x

Now that we have the value of b in terms of x, we can substitute it into the equation for x:

x = (16 + b) / 2.

Substituting b = 2x - 16, we get:

x = (16 + (2x - 16)) / 2.

Simplifying, we have:

x = (2x) / 2.

Cancelling out the 2's, we find:

x = x.

This equation tells us that the value of x can be any real number. Therefore, the length of the bisector of the angle cannot be determined solely based on the given information.

In conclusion, the length of the bisector of the angle cannot be determined with the given information.