Question

ДУЖЕ ТЕРМІНОВО! ДАЮ 100БАЛІВ! У трикутнику АВС АВ=АС, ВС=12 см, площа трикутника 18 см кв. Через вершину А проведено до площини трикутника перпендикуляр ДА такий, що відрізок ДЕ має довжину 3 помножене на корінь з 2 см, точка Е - середина ВС. Знайдіть кут між прямою ДЕ та площиною трикутника.

Answer 1

Объяснение:

Найдем радиус окружности, вписанной в данный треугольник

r=S/p, где S - площадь треугольника, р - полупериметр треугольника

Высота данного треугольника ВN=< var > \sqrt{AB^{2}-AK^{2}}=\sqrt{169-25}=\sqrt{144}=12 < /var ><var>

AB

2

−AK

2

=

169−25

=

144

=12</var> см

S=1/2*AC*BN=1/2*10*12=60 см^2

p=18 см, r=60/18=10/3

Тогда высота треугольника МВК=BN-2r=12-20/3=16/3

S MBK/S ABC=(16/3)^2/12^2

S MBK=11 целых 23/27

Answer 2

To find the angle between line DE and the plane of triangle ABC, we can use the concept of dot product between two vectors. Let's break down the problem step by step.

Given information:

- Triangle ABC with AB = AC - BC = 12 cm - Area of triangle ABC = 18 cm² - DE = 3√2 cm

Step 1: Finding the length of AB and AC

Since AB = AC, we can denote their length as x. Using the formula for the area of a triangle, we can find x.

The area of a triangle can be calculated using the formula: Area = (1/2) * base * height

In this case, the base is BC and the height is AB or AC. So we have: 18 = (1/2) * 12 * x

Simplifying the equation: 36 = 12x x = 3 cm

Therefore, AB = AC = 3 cm.

Step 2: Finding the length of BC

We are given that BC = 12 cm.

Step 3: Finding the length of DE

We are given that DE = 3√2 cm.

Step 4: Finding the coordinates of points A, B, and C

Since AB = AC = 3 cm, we can assume that A is the origin (0, 0, 0) and B is (3, 0, 0) in a 3D coordinate system.

Using the distance formula, we can find the coordinates of C: BC = √[(x2 - x1)² + (y2 - y1)² + (z2 - z1)²] 12 = √[(x - 3)² + y² + z²]

Simplifying the equation: 144 = (x - 3)² + y² + z²

Since AB = AC, the coordinates of C can be (3, ±√6, ±√6). We choose the positive values to keep the triangle in the same plane.

Therefore, C is (3, √6, √6).

Step 5: Finding the coordinates of point D

We are given that DE is perpendicular to the plane of triangle ABC and E is the midpoint of BC. Therefore, D is the projection of A onto the plane of triangle ABC.

To find the coordinates of D, we can use the formula for the projection of a point onto a plane:

D = A - [(A - C) dot n] * n

Where n is the normal vector of the plane, which can be found by taking the cross product of vectors AB and AC.

AB = (3, 0, 0) AC = (3, √6, √6)

n = AB x AC = (0, 0, 3√6)

Substituting the values into the formula: D = (0, 0, 0) - [(0, 0, 0) dot (0, 0, 3√6)] * (0, 0, 3√6) = (0, 0, 0) - (0) * (0, 0, 3√6) = (0, 0, 0)

Therefore, D is (0, 0, 0).

Step 6: Finding the vector DE

We can find the vector DE by subtracting the coordinates of D from the coordinates of E:

DE = E - D = (3, √6, √6) - (0, 0, 0) = (3, √6, √6)

Step 7: Finding the dot product between DE and the normal vector of the plane

To find the angle between line DE and the plane of triangle ABC, we can take the dot product between DE and the normal vector of the plane.

DE dot n = (3, √6, √6) dot (0, 0, 3√6) = 0 + 0 + 3√6 * 3√6 = 54

Step 8: Finding the magnitudes of DE and the normal vector of the plane

The magnitude of a vector can be found using the formula: magnitude = √(x² + y² + z²).

The magnitude of DE = √[(3)² + (√6)² + (√6)²] = √[9 + 6 + 6] = √21

The magnitude of the normal vector of the plane = √[(0)² + (0)² + (3√6)²] = √[0 + 0 + 54] = √54

Step 9: Finding the angle between DE and the plane of triangle ABC

The angle between two vectors can be found using the formula: cosθ = (A dot B) / (|A| * |B|).

Substituting the values: cosθ = (DE dot n) / (|DE| * |n|) = 54 / (√21 * √54)

Simplifying the equation: cosθ = 54 / (√(21 * 54)) = 54 / (√1134) = 54 / 33.71 ≈ 1.60

To find the angle θ, we can take the inverse cosine (arccos) of cosθ:

θ ≈ arccos(1.60) ≈ 0.99 radians

Therefore, the angle between line DE and the plane of triangle ABC is approximately 0.99 radians.

Note: The above calculations are based on the given information and assumptions made. Please double-check the values and calculations for accuracy.