Question

Катеты прямоугольного треугольника относятся как 3:4 а его периметр равен 72 см вычислить радиус описанной окружности

Answer 1

Пусть 1 катет = 3x, тогда 2 = 4x
Найдём гипотенузу по теореме Пифагора: √(9x² + 16x²) = √(25x²) = 5x

3x + 4x + 5x = P треугольника
12x = 72
x = 6

R = половине гипотенузы, так как треугольник прямоугольный, то есть R = 2.5x = 2.5 * 6 = 15 см

Ответ: 15 см

Answer 2

Calculation of the Radius of the Circumscribed Circle

To calculate the radius of the circumscribed circle of a right-angled triangle with the ratio of its legs as 3:4 and a perimeter of 72 cm, we can use the following steps:

Step 1: Determine the lengths of the legs of the right-angled triangle. According to the given information, the ratio of the legs is 3:4. Let's assume the lengths of the legs are 3x and 4x, where x is a common factor. Therefore, the lengths of the legs are 3x and 4x.

Step 2: Calculate the perimeter of the triangle. The perimeter of a triangle is the sum of the lengths of its sides. In this case, the perimeter is given as 72 cm. So, we can set up the equation: 3x + 4x + hypotenuse = 72 cm

Step 3: Find the hypotenuse of the triangle. Since the triangle is a right-angled triangle, we can use the Pythagorean theorem to find the length of the hypotenuse. The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. Using the lengths of the legs (3x and 4x), we can write the equation: (3x)^2 + (4x)^2 = hypotenuse^2

Step 4: Solve the equations. Now we can solve the equations to find the value of x and the length of the hypotenuse.

Let's calculate the values:

Equation 1: 3x + 4x + hypotenuse = 72 cm Equation 2: (3x)^2 + (4x)^2 = hypotenuse^2

Simplifying Equation 1: 7x + hypotenuse = 72 cm

Simplifying Equation 2: 9x^2 + 16x^2 = hypotenuse^2 25x^2 = hypotenuse^2

Since the hypotenuse is common in both equations, we can equate them: 25x^2 = hypotenuse^2

Now we can substitute the value of hypotenuse from Equation 1 into Equation 2: 25x^2 = (72 cm - 7x)^2

Expanding and simplifying Equation 2: 25x^2 = 5184 cm^2 - 1008x + 49x^2

Combining like terms: -24x^2 + 1008x - 5184 = 0

Now we can solve this quadratic equation to find the value of x.

Using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = -24, b = 1008, and c = -5184.

Solving for x, we get two possible values: x = 6 and x = -36/7. Since the length of a side cannot be negative, we discard the negative value.

Therefore, x = 6.

Step 5: Calculate the length of the hypotenuse. Using the value of x, we can calculate the length of the hypotenuse: hypotenuse = 72 cm - 7x hypotenuse = 72 cm - 7(6) hypotenuse = 72 cm - 42 cm hypotenuse = 30 cm

Step 6: Calculate the radius of the circumscribed circle. The radius of the circumscribed circle of a right-angled triangle is equal to half the length of the hypotenuse.

Therefore, the radius of the circumscribed circle is: radius = 1/2 * hypotenuse radius = 1/2 * 30 cm radius = 15 cm

So, the radius of the circumscribed circle of the given right-angled triangle is 15 cm.

Please note that the calculations provided above are based on the given information and assumptions.