Question

Катеты АВС прям. имеют длину 60 и 80 см.Из С к пл. треугольника востановлен перпендикуляр СD равный 36 см.Угол наклонной DF к пл. т и Δ ,где DF перпендик. опуш. из т. D на прям АВ. Помогите решить срочно ! Желательно подробно

Answer 1

 CF перпендикулярна АВ, по теореме о трех перпендикулярах. Наклонная DF перпендикулярна АВ по условию, значит и её проекция CF тоже перпендикулярна АВ. НО  CF  также - высота прямоугольного треугольника, опущенная на гипотенузу.
Найдем площадь прямоугольного  треуголоьника двумя способами. Сначала как половина произведения катетов. Получим    1/2 на 60 на 80 =2400 кв см.
Гипотенуза Ав по теореме Пифагора √60²+80²= √3600+6400=√10 000= 100 см
Площадь треугольника равна половине произведения основания АВ на высоту CF. 
2400=1|2 (100)·CF.  СF= 2400:50=48.
DF = √36²+48²=√3600=60
Из треугольника DFC  найдем sin DFC= DC :DF= 36:60= 0,6
угол DFC = arcsin 0,6

Answer 2

Given Information:

- The lengths of the legs AB and AC of a right triangle ABC are 60 cm and 80 cm, respectively. - A perpendicular CD is drawn from point C to the hypotenuse AB, and its length is 36 cm. - The angle between the inclined line DF and the plane of triangle ABC is denoted as θ.

Solution:

To solve this problem, we can use the properties of similar triangles and the Pythagorean theorem.

1. Let's start by finding the length of the hypotenuse BC using the Pythagorean theorem: - The length of AB is 60 cm. - The length of AC is 80 cm. - Let's denote the length of BC as x. - According to the Pythagorean theorem, we have: AB^2 + AC^2 = BC^2. - Substituting the given values, we get: 60^2 + 80^2 = x^2. - Solving this equation, we find: x^2 = 3600 + 6400 = 10000. - Taking the square root of both sides, we get: x = 100 cm.

2. Now, let's find the length of the perpendicular CD: - The length of CD is given as 36 cm.

3. Next, let's find the length of the inclined line DF: - Let's denote the length of DF as y. - Since triangle CDF is a right triangle, we can use the Pythagorean theorem to find y. - According to the Pythagorean theorem, we have: CD^2 + DF^2 = CF^2. - Substituting the given values, we get: 36^2 + y^2 = 100^2. - Solving this equation, we find: y^2 = 10000 - 1296 = 8704. - Taking the square root of both sides, we get: y = 93.28 cm (rounded to two decimal places).

4. Finally, let's find the angle θ between the inclined line DF and the plane of triangle ABC: - We can use the tangent function to find the angle θ. - The tangent of θ is given by the ratio of the length of DF to the length of AB. - Substituting the values, we have: tan(θ) = DF / AB = 93.28 / 60. - Using a calculator, we find: θ ≈ 56.31 degrees (rounded to two decimal places).

Summary:

In summary, for the given right triangle ABC with leg lengths of 60 cm and 80 cm, the length of the hypotenuse BC is 100 cm. The length of the perpendicular CD is 36 cm. The length of the inclined line DF is approximately 93.28 cm. The angle θ between the inclined line DF and the plane of triangle ABC is approximately 56.31 degrees.