Question

Помогите пожалуйста с задачкой:) Гипотенуза прямоугольного треугольника равна 17см, а периметр равен 40см Найдите катеты прямоугольного треугольника РЕШИТЬ ТАК, ЧТОБЫ ПОЛУЧИЛАСЬ СИСТЕМА УРАВНЕНИЙ

Answer 1

Problem Analysis

We are given that the hypotenuse of a right triangle is 17 cm and the perimeter is 40 cm. We need to find the lengths of the two legs of the triangle.

Solution

Let's assume that the two legs of the right triangle are represented by variables x and y. We can set up the following system of equations based on the given information:

1. The Pythagorean theorem: x^2 + y^2 = 17^2 (Equation 1) 2. The perimeter equation: x + y + 17 = 40 (Equation 2)

We can solve this system of equations to find the values of x and y.

Solving the System of Equations

To solve the system of equations, we can use the substitution method or the elimination method. Let's use the substitution method.

From Equation 2, we can isolate y: y = 40 - x - 17

Substituting this value of y into Equation 1, we get: x^2 + (40 - x - 17)^2 = 17^2

Simplifying the equation: x^2 + (23 - x)^2 = 289

Expanding and rearranging the equation: x^2 + 529 - 46x + x^2 = 289

Combining like terms: 2x^2 - 46x + 240 = 0

Now we have a quadratic equation. We can solve it by factoring or using the quadratic formula. Let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this equation, a = 2, b = -46, and c = 240.

Substituting the values into the quadratic formula: x = (-(-46) ± √((-46)^2 - 4 * 2 * 240)) / (2 * 2)

Simplifying: x = (46 ± √(2116 - 1920)) / 4

Calculating the discriminant: x = (46 ± √196) / 4

Simplifying further: x = (46 ± 14) / 4

We have two possible solutions for x: 1. x = (46 + 14) / 4 = 60 / 4 = 15 2. x = (46 - 14) / 4 = 32 / 4 = 8

Now, we can substitute these values of x back into Equation 2 to find the corresponding values of y.

For x = 15: y = 40 - 15 - 17 = 8

For x = 8: y = 40 - 8 - 17 = 15

Therefore, the lengths of the two legs of the right triangle are: 1. x = 15 cm 2. y = 8 cm

Answer

The lengths of the two legs of the right triangle are 15 cm and 8 cm.