Question

Периметр прямоугольного треугольника равен 24 см,а его гипотенуза 10 см.найдите катеты треугольника 9 класс,помогите)

Answer 1

1) 24-10=14
отсюда по правилу катеты прямоугольного треугольника равны
2) 14:2=7 значит катеты равны 7 см

Answer 2

Finding the Length of the Legs of a Right Triangle

To find the lengths of the legs of a right triangle given the perimeter and the length of the hypotenuse, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

Let's denote the lengths of the legs as a and b, and the length of the hypotenuse as c. The perimeter of the triangle is the sum of the lengths of all three sides, which in this case is given as 24 cm. Therefore, we have the equation:

a + b + c = 24 (Equation 1)

We are also given that the length of the hypotenuse is 10 cm:

c = 10 (Equation 2)

To find the lengths of the legs, we need to solve these two equations simultaneously.

Solving the Equations

Substituting Equation 2 into Equation 1, we get:

a + b + 10 = 24

Rearranging the equation, we have:

a + b = 14 (Equation 3)

Now, we have two equations:

a + b = 14 (Equation 3)

c = 10 (Equation 2)

To solve for the lengths of the legs, we need one more equation. Unfortunately, the given information does not provide us with enough information to find the exact values of a and b. However, we can still express one of the variables in terms of the other.

Expressing One Variable in Terms of the Other

Let's express a in terms of b using Equation 3:

a = 14 - b

Now, we can substitute this expression for a into Equation 2 to find the value of b:

c = 10

10 = sqrt((14 - b)^2 + b^2) (Equation 4)

Simplifying Equation 4, we have:

100 = (14 - b)^2 + b^2

Expanding and rearranging the equation, we get:

100 = 196 - 28b + b^2 + b^2

Combining like terms, we have:

2b^2 - 28b + 96 = 0

Now, we can solve this quadratic equation to find the possible values of b.

Solving the Quadratic Equation

Using the quadratic formula, which states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:

x = (-b ± sqrt(b^2 - 4ac)) / (2a)

For our equation 2b^2 - 28b + 96 = 0, we have a = 2, b = -28, and c = 96. Substituting these values into the quadratic formula, we get:

b = (-(-28) ± sqrt((-28)^2 - 4(2)(96))) / (2(2))

Simplifying further, we have:

b = (28 ± sqrt(784 - 768)) / 4

b = (28 ± sqrt(16)) / 4

b = (28 ± 4) / 4

This gives us two possible values for b:

b = 8 or b = 6

Now, we can substitute these values of b back into Equation 3 to find the corresponding values of a:

For b = 8:

a = 14 - 8 = 6

For b = 6:

a = 14 - 6 = 8

Therefore, the possible lengths of the legs of the triangle are a = 6 cm and b = 8 cm, or a = 8 cm and b = 6 cm.

Conclusion

In conclusion, the lengths of the legs of the right triangle are either 6 cm and 8 cm, or 8 cm and 6 cm.