Question

Катети прямокутного трикутника відносяться як 5:12,а його медіана,проведена до гіпотенузи дорівнює 26 см. Знайти Р∆.

Answer 1

Обозначим катеты 5х и 12 х.
В прямоугольном треугольнике гипотенуза равна двум медианам, то есть 52 см.
По Пифагору 52² = (5х)²+(12х)²
2704 = 25х² + 144х²,
2704 = 169х²,
х = √(2704/169) = √16 = 4 см.
Тогда катеты равны 5*4 = 20 см и 12*4 = 48 см.
Периметр Р = 20+48+52 = 120 см.

Answer 2

Problem Analysis

We are given that the ratio of the legs of a right triangle is 5:12, and the length of the median drawn to the hypotenuse is 26 cm. We need to find the perimeter of the triangle.

Solution

Let's assume that the lengths of the legs of the right triangle are 5x and 12x, where x is a common factor.

According to the properties of a right triangle, the length of the median drawn to the hypotenuse is equal to half the length of the hypotenuse. Therefore, the length of the hypotenuse is 2 * 26 cm = 52 cm.

Using the Pythagorean theorem, we can find the value of x:

Leg 1 squared + Leg 2 squared = Hypotenuse squared

(5x)^2 + (12x)^2 = 52^2

25x^2 + 144x^2 = 2704

169x^2 = 2704

x^2 = 2704 / 169

x^2 = 16

x = 4

Now that we have the value of x, we can find the lengths of the legs:

Leg 1 = 5x = 5 * 4 = 20 cm

Leg 2 = 12x = 12 * 4 = 48 cm

The perimeter of the triangle is the sum of the lengths of all three sides:

Perimeter = Leg 1 + Leg 2 + Hypotenuse

Perimeter = 20 cm + 48 cm + 52 cm = 120 cm

Therefore, the perimeter of the triangle is 120 cm.

Conclusion

The perimeter of the right triangle with leg lengths in the ratio of 5:12 and a median drawn to the hypotenuse of 26 cm is 120 cm.