Use the distance formula to prove geometric properties and find equidistant points.
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Fig 7.6 showing students seated in a classroom.
photorealistic educational scene, shot on 35mm lens, natural lighting with soft bokeh, warm inviting color grading, gold…
Table mapping geometric conditions to coordinate geometry distance checks.
Worked example showing how to prove points form a square.
Problem. Show that the points , , and are the vertices of a square.
Proving Ashima, Bharti, and Camella are seated in a line.
To prove points A(3, 1), B(6, 4), and C(8, 6) are collinear, we must find the distances between them. Distance AB is calculated as and BC is calculated as . The total distance AC is , which evaluates to . Factoring out a perfect square, simplifies to . When we add AB and BC, we get . This technique of verifying distance sums is a core tool in coordinate geometry. Because the sum of the two shorter distances exactly equals the longest distance , we can conclude the three students are seated in a straight .
Setting up equations for points at equal distances.
When a point is equidistant from two other points and , it means the distance from to is exactly the same as the distance from to .
Geometrically, this means lies somewhere on the perpendicular bisector of the line segment joining and .
Worked example finding relation between x and y for an equidistant point.
Problem. Find a relation between and such that the point is equidistant from the points and .
Worked example finding a specific point on the y-axis.
Problem. Find a point on the y-axis which is equidistant from the points and .