Question

В трикутник АВС вписано коло, яке ділить одну із сторін трикутника на відрізки 7 та 6, друга сторона трикутника дорівнює 11. Знайти периметр трикутника.

Answer 1

Ответ:

Объяснение:

Відрізки дотичних до кола,проведені з однієї точки,рівні.На цій теоремі грунтується рішення.

Answer 2

Problem Analysis

We are given a triangle ABC with an inscribed circle. One of the sides of the triangle is divided by the circle into segments of length 7 and 6. The length of the second side of the triangle is 11. 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.

Let's denote the points where the circle intersects the side of the triangle as D and E. We can see that AD = 7 and AE = 6.

Since the circle is inscribed in the triangle, we know that the lengths of the tangents drawn from a point to a circle are equal. Therefore, BD = CE.

Let's denote the length of BD (or CE) as x.

We can now calculate the length of the third side of the triangle, BC, using the fact that the sum of the lengths of two sides of a triangle is greater than the length of the third side. In this case, we have:

BC + BD > CD BC + x > 11

We also know that the sum of the lengths of the tangents drawn from an external point to a circle is equal to the length of the third side of the triangle. Therefore, we have:

BD + CE = BC x + x = BC 2x = BC

Now we can substitute the value of BC in the previous inequality:

2x > 11 - x 3x > 11 x > 11/3

Since x represents the length of a line segment, it cannot be negative. Therefore, we have:

x > 11/3

Now we can calculate the perimeter of the triangle:

Perimeter = AB + BC + AC Perimeter = 7 + 2x + 6 Perimeter = 13 + 2x

Substituting the value of x, we have:

Perimeter > 13 + 2(11/3) Perimeter > 13 + 22/3 Perimeter > 59/3

Therefore, the perimeter of the triangle is greater than 59/3.

Answer

The perimeter of the triangle is greater than 59/3.