Question

В пра­виль­ной тре­уголь­ной пи­ра­ми­де SABC ме­ди­а­ны ос­но­ва­ния пе­ре­се­ка­ют­ся в точке O. Пло­щадь тре­уголь­ни­ка ABC равна 28, OS = 12. Най­ди­те объем пи­ра­ми­ды.

Answer 1

В основании - правильный треугольник. Медианы являются высотами и биссектрисами. Высота известна, площадь основания известна, находим объем пирамиды:
$V= \frac{1}{3}S*h=4*28=112$

Answer 2

Given Information:

We are given that in a right triangular pyramid SABC, the medians of the base intersect at point O. The area of triangle ABC is 28, and OS is equal to 12.

Solution:

To find the volume of the pyramid, we can use the formula:

Volume = (1/3) * Base Area * Height

To find the base area, we need to find the length of the base median. Let's denote the length of the base median as m.

Using the given information, we can find the length of the base median as follows:

Area of triangle ABC = (1/2) * Base * Height

Substituting the given values:

28 = (1/2) * Base * Height

Since the height is not given, we need to find it. Let's denote the height as h.

We can use the Pythagorean theorem to find the height:

h^2 = OS^2 - m^2

Substituting the given values:

h^2 = 12^2 - m^2

Now, we have two equations:

1. 28 = (1/2) * Base * h 2. h^2 = 12^2 - m^2

We can solve these equations simultaneously to find the values of Base and h.

Let's solve equation 2 for m:

m^2 = 12^2 - h^2

Substituting this value of m^2 in equation 1:

28 = (1/2) * Base * h

28 = (1/2) * Base * sqrt(12^2 - h^2)

Simplifying the equation:

56 = Base * sqrt(12^2 - h^2)

Now, we have an equation relating Base, h, and m.

To find the volume, we need to find the base area. The base area can be calculated using the formula for the area of a triangle:

Base Area = (1/2) * Base * m

Substituting the value of Base * m from the previous equation:

Base Area = (1/2) * (56 / sqrt(12^2 - h^2)) * m

Finally, substituting the value of the base area in the volume formula:

Volume = (1/3) * Base Area * Height

Volume = (1/3) * (1/2) * (56 / sqrt(12^2 - h^2)) * m * h

Simplifying the equation:

Volume = (28 / 3) * (m * h) / sqrt(12^2 - h^2)

Now, we have the formula to calculate the volume of the pyramid in terms of m and h.

Please note that the exact values of Base, h, and m cannot be determined without additional information or measurements.

Let me know if you need any further assistance!