Question

Катети прямокутного трикутника дорівнюють 12,5 см.Знайти периметр,площу,радіуси кіл,медіану,висоту проведену до гіпотенузи.

Answer 1

Катеты прямоугольного треугольника равны 12,5- следовательно, этот треугольник равнобедренный.
1)
Острые углы равнобедренного прямоугольного треугольника равны

90°:2=45° ( третий угол 90°)
2)
гипотенуза такого треугольника равна
с=√(2а²) =а√2=12,5√2 см или ≈ 17,677 см
3)
периметр
Р=12,5√2+2*12,5=12,5(√2+2) см или  ≈ 42,677 см
4)
площадь
S=а²:2=78,125 см²
5)
высота такого треугольника равна его медиане и равна половине гипотенузы
h=12,5√2:2=6√2 см или  ≈ 8,485 см

Answer 2

Given Information:

The lengths of the legs of a right triangle are 12.5 cm.

Perimeter of the Triangle:

The perimeter of a triangle is the sum of the lengths of its sides. In this case, the right triangle has two legs and a hypotenuse. To find the perimeter, we add the lengths of the legs and the hypotenuse:

Perimeter = leg1 + leg2 + hypotenuse

Since the lengths of the legs are given as 12.5 cm, the perimeter can be calculated as:

Perimeter = 12.5 cm + 12.5 cm + hypotenuse

To find the hypotenuse, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs:

hypotenuse^2 = leg1^2 + leg2^2

Let's calculate the hypotenuse using this formula:

hypotenuse^2 = 12.5 cm^2 + 12.5 cm^2 hypotenuse^2 = 156.25 cm^2 + 156.25 cm^2 hypotenuse^2 = 312.5 cm^2 hypotenuse = √312.5 cm hypotenuse ≈ 17.68 cm

Now we can calculate the perimeter:

Perimeter = 12.5 cm + 12.5 cm + 17.68 cm Perimeter ≈ 42.68 cm

Therefore, the perimeter of the right triangle is approximately 42.68 cm.

Area of the Triangle:

The area of a triangle can be calculated using the formula:

Area = (base * height) / 2

In a right triangle, the legs are the base and height. Therefore, the area can be calculated as:

Area = (leg1 * leg2) / 2

Substituting the given values:

Area = (12.5 cm * 12.5 cm) / 2 Area = 156.25 cm^2 / 2 Area ≈ 78.13 cm^2

Therefore, the area of the right triangle is approximately 78.13 cm^2.

Radii of the Circles:

To find the radii of the circles, we need to consider the inscribed and circumscribed circles of the right triangle.

1. Inscribed Circle: The radius of the inscribed circle can be calculated using the formula:

Radius = Area / Semiperimeter

The semiperimeter is half of the perimeter. Substituting the values:

Semiperimeter = Perimeter / 2 Semiperimeter = 42.68 cm / 2 Semiperimeter ≈ 21.34 cm

Radius = 78.13 cm^2 / 21.34 cm Radius ≈ 3.66 cm

Therefore, the radius of the inscribed circle is approximately 3.66 cm.

2. Circumscribed Circle: The radius of the circumscribed circle can be calculated using the formula:

Radius = Hypotenuse / 2

Substituting the value of the hypotenuse:

Radius = 17.68 cm / 2 Radius ≈ 8.84 cm

Therefore, the radius of the circumscribed circle is approximately 8.84 cm.

Median and Height:

In a right triangle, the median drawn from the right angle to the hypotenuse is equal to half the length of the hypotenuse. Similarly, the height drawn from the right angle to the hypotenuse is equal to half the product of the lengths of the legs divided by the hypotenuse.

1. Median: Median = Hypotenuse / 2 Median = 17.68 cm / 2 Median ≈ 8.84 cm

Therefore, the length of the median is approximately 8.84 cm.

2. Height: Height = (Leg1 * Leg2) / Hypotenuse Height = (12.5 cm * 12.5 cm) / 17.68 cm Height = 156.25 cm^2 / 17.68 cm Height ≈ 8.82 cm

Therefore, the length of the height is approximately 8.82 cm.

To summarize: - Perimeter ≈ 42.68 cm - Area ≈ 78.13 cm^2 - Radius of the inscribed circle ≈ 3.66 cm - Radius of the circumscribed circle ≈ 8.84 cm - Median ≈ 8.84 cm - Height ≈ 8.82 cm