Question

В правильной треугольной пирамиде SABC R-середина ребра AB, S-вершина .BC=4,а площадь боковой поверхности равна 36.Найти SR

Answer 1

В правильной треугольной пирамиде SR - это апофема (А).

Sбок = 1/2*(р*А), где р - периметр основания.

Отсюда А =2*Sбок / р = 2*36 / 3*4 = 6.

Answer 2

Sбок=1/2*(p*Апофема)
=>Апофема=(2sбок)/p=72/12=6

Answer 3

Problem Analysis

We are given a right triangular pyramid SABC, where R is the midpoint of edge AB, S is the vertex, BC = 4, and the lateral surface area is 36. We need to find SR.

Solution

To find SR, we need to find the length of the slant height of the triangular pyramid. Let's break down the solution into steps:

1. Find the area of the triangular face ABC. 2. Use the area of the triangular face to find the length of the slant height. 3. Use the length of the slant height to find SR.

Step 1: Find the Area of the Triangular Face ABC

To find the area of the triangular face ABC, we can use the formula for the area of a triangle: Area = (1/2) * base * height.

In this case, the base of the triangle is BC, which is given as 4. We need to find the height of the triangle.

Let's denote the height of the triangle as h. Since R is the midpoint of AB, we can use the Pythagorean theorem to find the height.

According to the Pythagorean theorem, the square of the hypotenuse (BC) is equal to the sum of the squares of the other two sides (BR and RC).

Using this information, we can write the equation: BC^2 = BR^2 + RC^2.

Since BC = 4 and BR = RC (as R is the midpoint of AB), we can rewrite the equation as: 4^2 = BR^2 + BR^2.

Simplifying the equation, we get: 16 = 2 * BR^2.

Solving for BR, we find: BR = sqrt(8).

Now that we have the base and height of the triangle, we can calculate the area using the formula: Area = (1/2) * base * height.

Substituting the values, we get: Area = (1/2) * 4 * sqrt(8).

Simplifying, we find: Area = 2 * sqrt(8).

Therefore, the area of the triangular face ABC is 2 * sqrt(8).

Step 2: Use the Area of the Triangular Face to Find the Length of the Slant Height

The lateral surface area of the pyramid is given as 36. This includes the area of the triangular face ABC, which we found in Step 1.

To find the area of the remaining three triangular faces, we can subtract the area of ABC from the total lateral surface area.

Let's denote the area of the remaining three triangular faces as A.

We know that the total lateral surface area is 36, and the area of ABC is 2 * sqrt(8).

Therefore, we can write the equation: 36 = 2 * sqrt(8) + A.

Simplifying the equation, we find: A = 36 - 2 * sqrt(8).

Now, we have the area of the remaining three triangular faces.

Step 3: Use the Length of the Slant Height to Find SR

The slant height of a triangular pyramid is the height of one of the triangular faces.

To find the length of the slant height, we can use the formula for the area of a triangle: Area = (1/2) * base * height.

In this case, the base of the triangle is BC, which is given as 4. We need to find the height of the triangle.

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

Using the formula for the area of a triangle, we can write the equation: A = (1/2) * BC * h.

Substituting the values, we get: 36 - 2 * sqrt(8) = (1/2) * 4 * h.

Simplifying the equation, we find: 36 - 2 * sqrt(8) = 2h.

Dividing both sides of the equation by 2, we get: 18 - sqrt(8) = h.

Therefore, the height of the triangle (which is also the slant height) is 18 - sqrt(8).

Finally, we need to find SR. Since R is the midpoint of AB, SR is half the slant height.

Therefore, SR = (18 - sqrt(8)) / 2.

Answer

The length of SR in the right triangular pyramid SABC is (18 - sqrt(8)) / 2.

Please note that the above solution is based on the given information and assumptions made. If there are any additional constraints or information, the solution may vary.