Question

В правильной четырёхугольной пирамиде SABCD точка О-центр основания, S-вершина, SO=12, AC=18. Найдите боковое ребро SB

Answer 1

Ребро = √(Высота² + (Диагональ/2)²)
SB = √(12² + (18/2)²) = √(144 + 81) = √225 = 15

Answer 2

Problem Analysis

We are given a right quadrilateral pyramid SABCD, where O is the center of the base, S is the vertex, SO = 12, and AC = 18. We need to find the length of the lateral edge SB.

Solution

To find the length of the lateral edge SB, we can use the Pythagorean theorem in triangle SBO.

Let's denote the length of SB as x. Since the pyramid is right, we can use the Pythagorean theorem to find the length of SB:

SB^2 = SO^2 + OB^2

To find OB, we need to find the length of OC first. Since O is the center of the base, OC is the perpendicular bisector of AC. Therefore, OC = AC/2 = 18/2 = 9.

Now, we can find OB using the Pythagorean theorem in triangle OBC:

OB^2 = OC^2 + BC^2

To find BC, we can use the fact that ABCD is a right quadrilateral. In a right quadrilateral, the diagonals are perpendicular and bisect each other. Therefore, BC = AC/√2 = 18/√2.

Now, we can substitute the values into the equations to find the length of SB:

OB^2 = OC^2 + BC^2 = 9^2 + (18/√2)^2

SB^2 = SO^2 + OB^2 = 12^2 + (9^2 + (18/√2)^2)

Finally, we can find the length of SB by taking the square root of SB^2:

SB = √(12^2 + (9^2 + (18/√2)^2))

Let's calculate the value of SB.

Calculation

Substituting the values into the equation:

SB = √(12^2 + (9^2 + (18/√2)^2))

SB = √(144 + (81 + 162))

SB = √(144 + 243)

SB = √387

Calculating the square root of 387:

SB ≈ 19.67

Therefore, the length of the lateral edge SB is approximately 19.67.

Answer

The length of the lateral edge SB in the given right quadrilateral pyramid SABCD is approximately 19.67.