Question

периметр прямокутного трикутника дорівнює 30 см а гіпотенуза 13 см.знайти катети трикутника

Answer 1

Пусть один катет будет равен x см, а другой - y см.

Если искать по теореме Пифагора, то: 

$x^2+y^2=13^2\\ x^2+y^2=169$

Если искать по формуле периметра, то:

x+y+13=30

x+y=30-13

x+y=17

У нас есть два уравнения, можно составить систему:

$\left \{ {{x+y=17} \atop {x^2+y^2=169}} \right.\\ \left \{ {{x=17-y} \atop {x^2+y^2=169}} \right\\ (17-y)^2+y^2=169\\ 289-34y+y^2+y^2=169\\ 2y^2-34y+120=0|:(2)\\ y^2-17y+60=0\\ D=289-240=49\\ y_1=\frac{17+7}{2}=12\\ y_2=\frac{17-7}{2}=5\\ x_1=17-12=5\\ x_2=17-5=12$

Катеты треугольника - 12 см и 5 см.

 

Answer 2

Perimeter of a Right Triangle

To find the lengths of the legs of a right triangle when the perimeter is 30 cm and the hypotenuse is 13 cm, we can use the Pythagorean theorem and the formula for the perimeter of a triangle.

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. In this case, we have the hypotenuse as 13 cm.

Let's assume the lengths of the legs are a and b. According to the Pythagorean theorem, we have:

a^2 + b^2 = 13^2

The formula for the perimeter of a triangle is:

Perimeter = a + b + c

Given that the perimeter is 30 cm, we can substitute the values into the formula:

30 = a + b + 13

Now we have a system of two equations:

a^2 + b^2 = 13^2 a + b + 13 = 30

We can solve this system of equations to find the lengths of the legs.

Let's solve the second equation for a:

a = 30 - b - 13

Substituting this value of a into the first equation:

(30 - b - 13)^2 + b^2 = 13^2

Expanding and simplifying:

900 - 60b + b^2 - 390 + 26b + b^2 = 169

Combining like terms:

2b^2 - 34b + 341 = 0

Now we can solve this quadratic equation for b using the quadratic formula:

b = (-(-34) ± √((-34)^2 - 4(2)(341))) / (2(2))

Simplifying:

b = (34 ± √(1156 - 2736)) / 4

b = (34 ± √(-1580)) / 4

Since the discriminant is negative, there are no real solutions for b. This means that there is no right triangle with a perimeter of 30 cm and a hypotenuse of 13 cm.

Therefore, it is not possible to find the lengths of the legs of the triangle given the provided information.