Geometry

Step 1: Use properties of the equilateral triangle.
The relationships between the inradius, circumradius, and exradius are well-known for an equilateral triangle and can be derived based on the side length.
Step 2: Conclusion.
Thus, the ratio is .
Final Answer:

Step 1: Calculate the Normal Vectors.
To find the angle between the two faces, first calculate the normal vectors of the planes formed by the points. The dot product formula can be used to find the angle.
Step 2: Conclusion.
The correct answer is (B), .

Step 1: Use the property of angles in a quadrilateral.
The sum of angles in a quadrilateral is :
Step 2: Express and in terms of each other.
Recall that .
Step 3: Simplify the given expression.
Step 4: Determine the correct answer.
The expression simplifies to .

Step 1: Understanding the Problem
We are given four points in 3D space: A(4, -2, 1), B(7, -4, 7), C(2, -5, 10), and D(-1, -3, 4). We need to determine the type of figure formed by these points.
Step 2: Calculating the Vectors
Calculate the vectors for the sides:
Step 3: Checking for Parallelogram
A quadrilateral is a parallelogram if opposite sides are equal and parallel.
Here, and , which indicates that opposite sides are equal and parallel.
Step 4: Verifying Other Options
Tetrahedron: A tetrahedron has four triangular faces, which is not the case here.
Rhombus: A rhombus has all sides equal, which is not verified here.
Square: A square has all sides equal and all angles 90 degrees, which is not verified here.
Step 5: Matching with the Options
The figure formed by the points is a parallelogram, which corresponds to option (B). Final Answer: The points form a (B) Parallelogram.

Let the coordinates of the foot of the perpendicular from the point to the given line be . Step 1: Find the slope of AP The slope of line is: The slope of line (joining and ) is: Since is perpendicular to the given line, their slopes satisfy: Solving for :
Step 2: Use the equation of the given line Since lies on the given line , we substitute into the equation:
Step 3: Solve for and Equating the two expressions for : Substituting into : Thus, the coordinates of the foot of the perpendicular are: which corresponds to option (b).

Step 1: Use the formula for the area of a triangle
The area of a triangle is given by: Step 2: Substitute the given values
Given:
- Base = ,
- Height = .
Substitute these values into the formula: Answer: Therefore, the area of the triangle is . So, the correct answer is option (1).