Geometry Exercise 7.1

Distance Calculations

• Distance between points (2, 3) and (4, 1) can be calculated using the distance formula: (d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}).
• For points (-5, 7) and (-1, 3), apply the same distance formula to find the result.
• For general points (a, b) and (-a, -b), the formula leads to a consistent relation regardless of specific values.

Specific Distance Example

• The distance between (0, 0) and (36, 15) can be determined using the distance formula, yielding a specific numerical result.
• This distance can then be compared to the distance between two towns A and B, analyzing if known values apply.

Collinearity Analysis

• Determining if points (1, 5), (2, 3), and (-2, -11) are collinear requires checking if the slopes between any two pairs of points are equal.

Triangle Type Check

• To verify if points (5, -2), (6, 4), and (7, -2) form an isosceles triangle, calculate the distances between each pair and assess if at least two distances are equal.

Quadrilateral Shape Confirmation

• Assess points A, B, C, and D for whether they form a square by checking the distances between adjacent points and the diagonals.
• Evaluation of points (-1, -2), (1, 0), (-1, 2), and (-3, 0) to determine if they form a specific quadrilateral, justifying the claims with distance measures.
• Repeat for points (-3, 5), (3, 1), (0, 3), (-1, -4) and (4, 5), (7, 6), (4, 3), (1, 2) to classify the shape.

Equidistant Point on x-axis

• Finding a point on the x-axis that maintains equal distance from (2, -5) and (-2, 9) involves setting up the equation using the distance formula, leading to a solution for the x-coordinate.

Distance Constraints

• Establishing values of y that maintain a fixed distance of 10 units between points P(2, -3) and Q(10, y) requires solving the related distance equation for y.