Question

В прямоугольном треугольнике АВС ( угол С =90 , АВ = 10 , АС= СВ+2) проведена биссектрисса СМ . найдите катеты треугольника АВС и радиус окружности , описанной около треугольника СМВ.

Answer 1

Пусть СВ=х; тогда АС=х+2; это катеты; по т.Пифогора
x^2+(x+2)^2=AB^2,
x^2+x^2+4x+4=100,
2x^2+4x-96=0,
x^2+2x-48=0,
D=(b/2)^2-ac=1+48=49,
x1=-1-7<0 - посторонний корень,
х2=-1+7=6 - катет ВС,
6+2=8 - катет АС;
АВ - диаметр, т.к. вписанный прямой угол С может опираться только на диаметр; значит R=10/2=5 - это радиус окружности, описанной около тр-ка АВС.

Answer 2

Given Information:

We are given the following information about a right triangle ABC: - Angle C = 90 degrees - Side AB = 10 - Side AC = Side BC + 2

We are also told that the bisector SM is drawn in the triangle ABC.

Finding the Length of the Triangle Sides:

To find the lengths of the triangle sides, we can use the properties of a right triangle and the given information.

Let's denote the length of side BC as x.

Using the Pythagorean theorem, we can find the length of side BC: BC^2 = AB^2 - AC^2x^2 = 10^2 - (x + 2)^2

Expanding the equation: x^2 = 100 - (x^2 + 4x + 4) x^2 = 100 - x^2 - 4x - 4 2x^2 + 4x - 96 = 0

Simplifying the equation: x^2 + 2x - 48 = 0 x^2 + 8x - 6x - 48 = 0 x(x + 8) - 6(x + 8) = 0 (x - 6)(x + 8) = 0

Solving for x, we get two possible values: x = 6 or x = -8

Since the length of a side cannot be negative, we discard the negative value. Therefore, the length of side BC is x = 6.

Using the given information that side AC = BC + 2, we can find the length of side AC: AC = 6 + 2 = 8

Therefore, the lengths of the triangle sides are: AB = 10, BC = 6, AC = 8.

Finding the Radius of the Circumscribed Circle:

To find the radius of the circle circumscribed around triangle SMB, we can use the properties of a right triangle and the given information.

Let's denote the radius of the circumscribed circle as R.

The radius of the circumscribed circle is equal to half the length of the hypotenuse of the right triangle SMB.

Using the Pythagorean theorem, we can find the length of the hypotenuse SM: SM^2 = AM^2 + AS^2SM^2 = (AB/2)^2 + AC^2 SM^2 = (10/2)^2 + 8^2 SM^2 = 5^2 + 64 SM^2 = 25 + 64 SM^2 = 89

Therefore, the length of SM is the square root of 89: SM = sqrt(89)

The radius of the circumscribed circle is half the length of SM: R = SM/2 = sqrt(89)/2

Therefore, the radius of the circle circumscribed around triangle SMB is sqrt(89)/2.

Summary:

In the given right triangle ABC, with angle C = 90 degrees, AB = 10, and AC = BC + 2, we found the lengths of the triangle sides to be AB = 10, BC = 6, and AC = 8. We also found the radius of the circle circumscribed around triangle SMB to be sqrt(89)/2.

Please note that the sources provided were not directly used to answer the question, as the solution was derived using mathematical principles and the given information.