Question

Помогите срочно пожалуйста дам 40 баллов!!!!!!!!!!! Высота CK прямоугольного треугольника ABC , проведенная к гипотенузе,делит ее на отрезки длиной 9 см и 25 см Найдите катеты и периметр треугольника

Answer 1

Пусть ∆АВС прямоугольный ,<С=90°

, СК перпендикулярно АС . По условию вся гипотеза 9+25=34 . Один отрезок, например АК=9, другой ВК=25 . Тк "Катет, прямоугольного треугольника , есть среднее пропорциональное между проекцией этого катета и гипотенузой ", то получаем АС=√(АК*АВ)=√(9*34)=3√34 ; ВС=√(ВК*АВ)=√(25*34)=5√34 ; Р=5√34+3√34+34=8√34+34

Answer 2

Finding the Length of the Hypotenuse

To find the lengths of the legs and the perimeter of a right triangle, we first need to determine the length of the hypotenuse. The problem statement mentions that the height of the triangle CK divides the hypotenuse into two segments, measuring 9 cm and 25 cm.

Let's assume that the length of the hypotenuse is represented by 'x'. According to the problem, we have the following information:

- The length of one segment of the hypotenuse is 9 cm. - The length of the other segment of the hypotenuse is 25 cm.

To find the length of the hypotenuse, we can use the Pythagorean theorem, which 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.

Using the Pythagorean theorem, we can set up the following equation:

x^2 = 9^2 + 25^2

Simplifying the equation:

x^2 = 81 + 625

x^2 = 706

Taking the square root of both sides:

x = √706

Using a calculator, we find that the square root of 706 is approximately 26.57.

Therefore, the length of the hypotenuse (x) is approximately 26.57 cm.

Finding the Lengths of the Legs

Now that we know the length of the hypotenuse, we can find the lengths of the legs of the right triangle.

Let's label the legs of the triangle as 'a' and 'b'. According to the problem, the height of the triangle CK divides the hypotenuse into two segments, measuring 9 cm and 25 cm.

Using the concept of similar triangles, we can set up the following proportions:

a/9 = b/25

Cross-multiplying:

25a = 9b

Simplifying the equation:

a = (9b)/25

Now, we can substitute the value of 'a' in terms of 'b' into the Pythagorean theorem equation:

x^2 = a^2 + b^2

Substituting the value of 'a':

(26.57)^2 = [(9b)/25]^2 + b^2

Simplifying the equation:

706 = (81b^2)/625 + b^2

Multiplying both sides by 625 to eliminate the denominators:

706 * 625 = 81b^2 + 625b^2

Simplifying the equation:

706 * 625 = 706b^2

Dividing both sides by 706:

625 = b^2

Taking the square root of both sides:

b = √625

Simplifying the equation:

b = 25

Therefore, the length of one leg of the right triangle (b) is 25 cm. Substituting the value of 'b' into the equation for 'a':

a = (9 * 25)/25

Simplifying the equation:

a = 9

Therefore, the length of the other leg of the right triangle (a) is 9 cm.

Finding the Perimeter of the Triangle

To find the perimeter of the triangle, we need to add up the lengths of all three sides.

The lengths of the legs are 9 cm and 25 cm, and the length of the hypotenuse is approximately 26.57 cm.

Therefore, the perimeter of the triangle is:

Perimeter = 9 + 25 + 26.57

Calculating the sum:

Perimeter ≈ 60.57 cm

Therefore, the perimeter of the triangle is approximately 60.57 cm. In summary: - The length of the hypotenuse is approximately 26.57 cm. - The lengths of the legs are 9 cm and 25 cm. - The perimeter of the triangle is approximately 60.57 cm.