Question

25 баллов. Один из углов прямоугольного треугольника равен 30 градусам.А Медиана проведенная до гипотенузы равна 22 см. Задание 1 Найдите висоту треугольника. Задание 2 Найдите радиус вписанного круга. Надеюсь на отличный ответ и показ действий.

Answer 1

Ответ:

11√3 см.;  11√3 - 1 cм.

Объяснение:

Дано: ∠АВС,  ∠А=90°,  ∠В=30°,  АМ - медиана,  АМ=22 см.

Найти h,  r.

Медиана, проведенная к гипотенузе, равна половине гипотенузы. ВС=2 АМ=22*2=44 см.

АС=1/2 ВС=44:2=22 см по свойству катета, лежащего против угла 30°.

По теореме Пифагора АВ=√(ВС²-АС²)=√1936-484)=√1452=22√3 см.

ΔАСМ - равносторонний, АН - высота,  по формуле h=a√2/2

АН=22√3/2=11√3 см.

r=(a+b-c)/2=(22√3+22-44)/2=22(√3+1-2)/2=11√3 - 1 cм.

Answer 2

Task 1: Finding the height of the triangle

To find the height of the triangle, we can use the fact that the median of a triangle divides the triangle into two smaller triangles with equal areas.

Let's denote the height of the triangle as h.

Since the median divides the triangle into two smaller triangles with equal areas, we can set up the following equation:

Area of the triangle = (1/2) * base * height = (1/2) * (base of the triangle) * (height of the triangle)

Let's denote the base of the triangle as b.

We know that the median is drawn to the hypotenuse, which is the longest side of the right-angled triangle. Therefore, the base of the triangle is equal to twice the length of the median.

So, we have:

(1/2) * 2 * (median) * (height) = (1/2) * (b) * (h)

Simplifying the equation, we get:

median * height = (1/2) * b * h

Since we know the value of the median (22 cm) and the base (the hypotenuse), we can substitute these values into the equation and solve for the height.

Let's calculate the height using the given values:

median = 22 cm

base = hypotenuse

To find the base (hypotenuse), we can use trigonometric ratios. Since we know one angle of the triangle (30 degrees) and the length of the median, we can use the sine function to find the hypotenuse.

The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.

Let's denote the hypotenuse as c.

sin(30 degrees) = (opposite side) / (hypotenuse)

We know that the opposite side is equal to half the length of the median (since the median divides the hypotenuse into two equal parts).

So, we have:

sin(30 degrees) = (1/2) * median / c

Simplifying the equation, we get:

c = (1/2) * median / sin(30 degrees)

Now, we can substitute the values of the median and the angle into the equation to find the value of the hypotenuse (base).

Let's calculate the value of the hypotenuse (base):

median = 22 cm

angle = 30 degrees

c = (1/2) * 22 cm / sin(30 degrees)

Using a calculator, we can find that c ≈ 44.8 cm.

Now, we can substitute the values of the median and the base into the equation we derived earlier to find the height.

Let's calculate the height:

median = 22 cm

base = 44.8 cm

median * height = (1/2) * base * height

22 cm * height = (1/2) * 44.8 cm * height

Simplifying the equation, we get:

22 cm * height = 22.4 cm * height

Since the left side of the equation is equal to the right side, we can conclude that the height of the triangle is 22.4 cm.

Therefore, the answer to Task 1 is:

The height of the triangle is 22.4 cm.

Task 2: Finding the radius of the inscribed circle

To find the radius of the inscribed circle, we can use the fact that the radius of the inscribed circle is equal to the product of the area of the triangle and the semiperimeter of the triangle, divided by the area of the triangle.

Let's denote the radius of the inscribed circle as r.

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

Area of the triangle = (1/2) * base * height

We have already calculated the height of the triangle as 22.4 cm in Task 1.

To find the base of the triangle, we can use the value of the hypotenuse (base) we calculated earlier.

Let's denote the base of the triangle as b.

base = 44.8 cm

Now, we can calculate the area of the triangle:

Area of the triangle = (1/2) * base * height

Area of the triangle = (1/2) * 44.8 cm * 22.4 cm

Using a calculator, we can find that Area of the triangle ≈ 502.4 cm².

The semiperimeter of the triangle can be calculated by adding the lengths of all three sides of the triangle and dividing the sum by 2.

Let's denote the semiperimeter of the triangle as s.

Since we know the lengths of two sides of the triangle (the median and the base), we can calculate the length of the third side (the other leg of the triangle) using the Pythagorean theorem.

Let's denote the length of the third side as a.

median = 22 cm

base = 44.8 cm

Using the Pythagorean theorem, we have:

a² = c² - b²

a² = (44.8 cm)² - (22 cm)²

Using a calculator, we can find that a ≈ 39.2 cm.

Now, we can calculate the semiperimeter of the triangle:

s = (median + base + a) / 2

s = (22 cm + 44.8 cm + 39.2 cm) / 2

Using a calculator, we can find that s ≈ 53 cm.

Now, we can calculate the radius of the inscribed circle:

radius of the inscribed circle = (Area of the triangle * s) / (Area of the triangle)

radius of the inscribed circle = (502.4 cm² * 53 cm) / (502.4 cm²)

Simplifying the equation, we get:

radius of the inscribed circle = 53 cm

Therefore, the answer to Task 2 is:

The radius of the inscribed circle is 53 cm.

I hope this explanation helps! Let me know if you have any further questions.