Understanding the Distance Formula in Analytic Geometry

Jezriel Ortiz

12/21/2025

Pre-Calculus

Distance formula ( analytic Geom )

•

Dec 21, 2025

•

2 pages

Understanding the Distance Formula in Analytic Geometry

Jezriel Ortiz

@jezrielortiz

Coordinate geometry helps us work with points on a graph... Show more

Distance Between Two Points

Ever wondered how to measure the distance between two points on a graph? The distance formula makes this super easy: Distance = √ $(x₂ - x₁)² + (y₂ - y₁)²$

For example, finding the distance between A(2, 3) and B(10, 3) is straightforward:

Substitute the coordinates into the formula
Distance = √ $(10 - 2)² + (3 - 3)²$ = √ $64 + 0$ = 8 units

The distance formula can also help you find unknown coordinates. If you know that point A(-2, 5) and B(4, p) are √45 units apart, you can solve for p: √45 = √ $(4-(-2))² + (p-5)²$ √45 = √ $36 + (p-5)²$ This gives us p = 2 or p = 8

Pro Tip: When finding unknown coordinates, you'll often get two possible answers. Always check if both make sense in the context of the problem!

You can also use the distance formula to find points that are equidistant (the same distance) from two other points. This is super useful in more advanced geometry problems.

Solving Distance Problems

The distance formula becomes really powerful when working with geometric shapes. Let's see it in action with triangles and circles!

When finding the perimeter of a triangle, calculate the length of each side using the distance formula, then add them together. For example, with vertices A(6,0), B(2,3), and C(-1,-4), we find:

AB = 5 units
BC = √58 units
AC = √65 units
Perimeter = 5 + √58 + √65 units

For circle problems, remember that a diameter is twice the radius. When two points form a diameter, the distance between them equals 2r. In a problem where points A(-8,3) and B(h,8) form a diameter of a circle with radius 6.5 units:

The diameter is 13 units
Using the distance formula: 13 = √ $(h-(-8))² + (8-3)²$
Solving for h gives us h = 4 or h = -20

Remember: When drawing shapes on a coordinate plane, each vertex has specific coordinates that determine the shape's size and position. The distance formula helps you find lengths precisely!

Coordinate geometry connects algebra and geometry, giving you powerful tools to analyze shapes without having to measure them physically.

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Pre-Calculus

•

Dec 21, 2025

•

2 pages

Understanding the Distance Formula in Analytic Geometry

Jezriel Ortiz

@jezrielortiz

Coordinate geometry helps us work with points on a graph using math formulas. In this section, we'll explore how to calculate distances between points and solve problems using the distance formula, which is essential for understanding shapes and locations in... Show more

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Distance Between Two Points

Ever wondered how to measure the distance between two points on a graph? The distance formula makes this super easy: Distance = √ $(x₂ - x₁)² + (y₂ - y₁)²$

For example, finding the distance between A(2, 3) and B(10, 3) is straightforward:

Substitute the coordinates into the formula
Distance = √ $(10 - 2)² + (3 - 3)²$ = √ $64 + 0$ = 8 units

Pro Tip: When finding unknown coordinates, you'll often get two possible answers. Always check if both make sense in the context of the problem!

You can also use the distance formula to find points that are equidistant (the same distance) from two other points. This is super useful in more advanced geometry problems.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Solving Distance Problems

The distance formula becomes really powerful when working with geometric shapes. Let's see it in action with triangles and circles!

When finding the perimeter of a triangle, calculate the length of each side using the distance formula, then add them together. For example, with vertices A(6,0), B(2,3), and C(-1,-4), we find:

AB = 5 units
BC = √58 units
AC = √65 units
Perimeter = 5 + √58 + √65 units

The diameter is 13 units
Using the distance formula: 13 = √ $(h-(-8))² + (8-3)²$
Solving for h gives us h = 4 or h = -20

Remember: When drawing shapes on a coordinate plane, each vertex has specific coordinates that determine the shape's size and position. The distance formula helps you find lengths precisely!

Coordinate geometry connects algebra and geometry, giving you powerful tools to analyze shapes without having to measure them physically.