Question

Бісектриса кута прямокутника ділить його діагональ на відрізки завдовжки 30 і 40 см , починаючи від ближчої до цього кута вершини. знайдіть довжину кола , вписаного в прямокутний трикутник . гіпотенузою якого є діагональ, а катетами-сторони даного прямокутника,

Answer 1

Пусть х и у-стороны прямоугольника.Тогда по свойству биссектрисы треугольника 30/х=40/у или 3/х=4/у.По т.Пифагора
х²+у²=70²,х=0,75у.
Тогда (0,75у)²+у²=70²,             0,5625у²+у²=4900,         1,5625у²=4900
у²=4900:1,5625=3136, у=56(см).
Тогда х=√4900-3136=√1754=42(см)
Тогда r=(a+b-c)/2=(56+42-70)/2=(98-70)/2=28/2=14(cм)

Answer 2

To find the length of the circle inscribed in a right triangle, we need to use the properties of the triangle and the given information. Let's break down the problem step by step.

Given Information:

- The diagonal of the rectangle is divided into two segments by the angle bisector of the rectangle. - The lengths of the two segments are 30 cm and 40 cm, starting from the vertex closest to the angle bisector.

Step 1: Finding the Lengths of the Sides of the Rectangle:

Since the diagonal of the rectangle is the hypotenuse of the right triangle, we can use the Pythagorean theorem to find the lengths of the sides of the rectangle.

Let's assume the sides of the rectangle are a and b, with a being the shorter side adjacent to the 30 cm segment and b being the longer side adjacent to the 40 cm segment.

Using the Pythagorean theorem, we have:

a^2 + b^2 = diagonal^2

Substituting the values, we get:

a^2 + b^2 = (30 cm + 40 cm)^2 = 70^2 cm^2

Step 2: Finding the Length of the Hypotenuse of the Right Triangle:

The diagonal of the rectangle is the hypotenuse of the right triangle. We can use the Pythagorean theorem again to find its length.

Let's assume the length of the hypotenuse is c.

Using the Pythagorean theorem, we have:

a^2 + b^2 = c^2

Substituting the values, we get:

70^2 cm^2 = c^2

Taking the square root of both sides, we find:

c = 70 cm

Therefore, the length of the hypotenuse of the right triangle is 70 cm.

Step 3: Finding the Length of the Circle Inscribed in the Right Triangle:

To find the length of the circle inscribed in the right triangle, we can use the formula:

radius = (a + b - c) / 2

Substituting the values, we get:

radius = (30 cm + 40 cm - 70 cm) / 2 = 0 cm

Since the radius is 0 cm, this means that the circle is degenerate and reduces to a single point at the incenter of the triangle.

Therefore, the length of the circle inscribed in the right triangle is 0 cm.

I hope this helps! Let me know if you have any further questions.