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Geometric Shapes & Equidistance

Use the distance formula to prove geometric properties and find equidistant points.

8 micro-lessons

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Identifying Quadrilaterals

Table mapping geometric conditions to coordinate geometry distance checks.

Parallelogram

Opposite sides are equal in length
Diagonals bisect each other
Diagonals are NOT necessarily equal

Rectangle

Opposite sides are equal in length
Diagonals bisect each other
Diagonals ARE equal

Rhombus

All four sides are equal in length
Diagonals bisect each other at 90°
Diagonals are NOT necessarily equal

Square

All four sides are equal in length
Diagonals bisect each other at 90°
Diagonals ARE equal

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Example 2: Proving a Square

Worked example showing how to prove points form a square.

Example 2: Proving a Square

Problem. Show that the points $A(1, 7)$ , $B(4, 2)$ , $C(-1, -1)$ and $D(-4, 4)$ are the vertices of a square.

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Classroom Seating

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…

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Faded Example: Collinearity

Proving Ashima, Bharti, and Camella are seated in a line.

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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 $3\sqrt{2}$ and BC is calculated as $2\sqrt{2}$ . The total distance AC is $\sqrt{(8-3)^2 + (6-1)^2}$ , which evaluates to $\sqrt{50}$ . Factoring out a perfect square, $\sqrt{50}$ simplifies to $\sqrt{2}$ . When we add AB and BC, we get $3\sqrt{2} + 2\sqrt{2} = 5\sqrt{2}$ . 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 .

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Understanding Equidistance

Setting up equations for points at equal distances.

What is Equidistance?

When a point $P(x, y)$ is equidistant from two other points $A$ and $B$ , it means the distance from $P$ to $A$ is exactly the same as the distance from $P$ to $B$ .

Geometrically, this means $P$ lies somewhere on the perpendicular bisector of the line segment joining $A$ and $B$ .

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Example 4: Equidistance Relation

Worked example finding relation between x and y for an equidistant point.

Example 4: Equidistance Relation

Problem. Find a relation between $x$ and $y$ such that the point $P(x, y)$ is equidistant from the points $A(7, 1)$ and $B(3, 5)$ .

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Example 5: Equidistant Point on an Axis

Worked example finding a specific point on the y-axis.

Example 5: Equidistant Point on an Axis

Problem. Find a point on the y-axis which is equidistant from the points $A(6, 5)$ and $B(-4, 3)$ .

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← previousThe Distance Formula next →The Section Formula

Example 2: Proving a Square

Worked example showing how to prove points form a square.

Example 2: Proving a Square

Problem. Show that the points $A(1, 7)$ , $B(4, 2)$ , $C(-1, -1)$ and $D(-4, 4)$ are the vertices of a square.

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Faded Example: Collinearity

Proving Ashima, Bharti, and Camella are seated in a line.

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Understanding Equidistance

Setting up equations for points at equal distances.

What is Equidistance?

When a point $P(x, y)$ is equidistant from two other points $A$ and $B$ , it means the distance from $P$ to $A$ is exactly the same as the distance from $P$ to $B$ .

Geometrically, this means $P$ lies somewhere on the perpendicular bisector of the line segment joining $A$ and $B$ .

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Example 4: Equidistance Relation

Worked example finding relation between x and y for an equidistant point.

Example 4: Equidistance Relation

Problem. Find a relation between $x$ and $y$ such that the point $P(x, y)$ is equidistant from the points $A(7, 1)$ and $B(3, 5)$ .

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Example 5: Equidistant Point on an Axis

Worked example finding a specific point on the y-axis.

Example 5: Equidistant Point on an Axis

Problem. Find a point on the y-axis which is equidistant from the points $A(6, 5)$ and $B(-4, 3)$ .

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