Question

В прямокутний трикутник вписано коло, точка дотику ділить гіпотенузу на відрізки 3 і 10. знайти катет.

Answer 1

Прямоугольный треугольник АВС, Точки Д, Е, Н - это точки касания вписанного круга со сторонами АВ, ВС и АС соответственно.
АД=3, ДВ=10
Т.к. отрезки касательных к окружности, проведенные из одной точки, равны, то значит:
АД=АН=3
ДВ=ЕВ=10
НС=СЕ
Значит катеты АС=АН+НС=3+НС
ВС=СЕ+ЕВ=НС+10
Т.к. АС²+ВС²=АВ²
(3+НС)²+(10+НС)²=(3+10)²
9+6НС+НС²+100+20НС+НС²=169
НС²+13НС-30=0
D=169+120=289=17²
НС=(-13+17)/2=2
Катеты АС=3+2=5 и ВС=10+2=12

Answer 2

Problem Analysis

We are given a right triangle inscribed in a circle, and the point of tangency divides the hypotenuse into segments of length 3 and 10. We need to find the length of one of the legs (the cathetus).

Solution

Let's denote the length of the cathetus as x.

According to the theorem of a tangent to a circle, the length of the segment from the point of tangency to the point of intersection of the tangent with the hypotenuse is equal to the geometric mean of the lengths of the two segments of the hypotenuse.

In this case, the length of the segment from the point of tangency to the point of intersection with the hypotenuse is 3, and the length of the other segment is 10. Therefore, we can set up the following equation:

x * x = 3 * 10

Simplifying the equation, we have:

x^2 = 30

Taking the square root of both sides, we find:

x = √30

So, the length of the cathetus is equal to the square root of 30.

Answer

The length of the cathetus is √30.

Verification

To verify the answer, we can substitute the value of x into the equation and check if it satisfies the given conditions.

Substituting √30 into the equation, we have:

(√30)^2 = 3 * 10

Simplifying the equation, we get:

30 = 30

Since both sides of the equation are equal, we can conclude that the answer is correct.

Conclusion

In a right triangle inscribed in a circle, if the point of tangency divides the hypotenuse into segments of length 3 and 10, then the length of one of the legs (the cathetus) is equal to the square root of 30.