Question

ГЕОМЕТИЯ 10 !!В прямой треугольной призме ABCA1B1C1, угол ACB=90 градусов, угол BAC=60 градусов, АС=а. Прямая B1C составляет с плоскостью грани АА1С1С угол 45 градусов. Найдите площадь боковой поверхности призмы."

Answer 1

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Answer 2

To find the lateral surface area of a right triangular prism, we need to know the dimensions of the prism. In this case, we are given that the prism is ABCA1B1C1, where angle ACB is 90 degrees, angle BAC is 60 degrees, and AC is represented by 'a'. We are also given that the line B1C forms a 45-degree angle with the plane of the face AA1C1C.

Finding the Lateral Surface Area of the Prism

To find the lateral surface area of the prism, we need to calculate the sum of the areas of all the lateral faces.

Let's break down the prism into its individual faces: - Face ABC is a right triangle with AC as the hypotenuse and angle ACB as the right angle. - Face A1B1C1 is a right triangle with A1C1 as the hypotenuse and angle A1C1B1 as the right angle. - Face B1C is a right triangle with BC as the hypotenuse and angle B1CB as the right angle.

To find the lateral surface area, we need to calculate the areas of these three faces and sum them up.

Calculating the Area of Face ABC

Since angle ACB is 90 degrees and angle BAC is 60 degrees, we can determine that angle BCA is 180 - 90 - 60 = 30 degrees. This means that triangle ABC is a 30-60-90 right triangle.

In a 30-60-90 right triangle, the ratio of the sides is 1 : √3 : 2. Since AC is represented by 'a', BC is √3 * a, and AB is 2 * a.

The area of a triangle can be calculated using the formula: area = (base * height) / 2. In this case, the base is AB and the height is BC.

Substituting the values, we have: - base = 2 * a - height = √3 * a

So, the area of face ABC is (2 * a * √3 * a) / 2 = a^2 * √3.

Calculating the Area of Face A1B1C1

Since angle A1C1B1 is 90 degrees and angle A1C1A is 45 degrees, we can determine that angle A1A is 180 - 90 - 45 = 45 degrees. This means that triangle A1A is an isosceles right triangle.

In an isosceles right triangle, the ratio of the sides is 1 : 1 : √2. Since A1C1 is represented by 'a', A1A is a, and A1B1 is √2 * a.

The area of face A1B1C1 can be calculated using the same formula as before: - base = A1B1 = √2 * a - height = A1C1 = a

So, the area of face A1B1C1 is (√2 * a * a) / 2 = a^2 * √2 / 2.

Calculating the Area of Face B1C

Since angle B1CB is 90 degrees and angle B1CA1 is 45 degrees, we can determine that angle B1A1 is 180 - 90 - 45 = 45 degrees. This means that triangle B1C is an isosceles right triangle.

Using the same reasoning as before, the area of face B1C is (√2 * a * a) / 2 = a^2 * √2 / 2.

Summing Up the Areas

To find the lateral surface area of the prism, we need to sum up the areas of the three faces: - Area of face ABC: a^2 * √3 - Area of face A1B1C1: a^2 * √2 / 2 - Area of face B1C: a^2 * √2 / 2

Adding these areas together, we get the lateral surface area of the prism: a^2 * √3 + a^2 * √2 / 2 + a^2 * √2 / 2 = a^2 * √3 + a^2 * √2

So, the lateral surface area of the given prism is a^2 * √3 + a^2 * √2.

Please note that this calculation assumes that the prism is a right triangular prism with the given angles and dimensions.