Chelsea

12/8/2025

Algebra 2

Distance and Mid Point

Subjects

Algebra 2

Modeling

Manipulating Formulas

•

Dec 8, 2025

•

2 pages

Understanding Distance and Midpoint

Chelsea

@zuncho

Ready to master coordinate geometry? In this guide, we'll explore... Show more

480/81
Day 1
Notes

# DISTANCE AND MIDPOINT NOTES

OBJECTIVE:
1) Calculate distance between two points and find the midpoint of a segment in

Distance and Midpoint Formulas

Ever wondered how to find the exact distance between two points on a map or graph? The distance formula gives us the answer: d = √ $(x₂-x₁)² + (y₂-y₁)²$ . This formula applies the Pythagorean theorem to coordinate geometry.

The midpoint formula helps you find the exact center point between two coordinates. Simply average the x-coordinates and average the y-coordinates: $x₁+x₂$ /2, $y₁+y₂$ /2. Think of it as finding the "halfway point" between two locations.

Let's see this in action with an example: For points (-3,4) and (5,2), the distance is calculated as d = √ $(-3-5)² + (4-2)²$ = √ $(−8)² + 4$ = √ $64+4$ = √68 = 2√17. The midpoint is ((−3+5)/2, (4+2)/2) = (1,3).

Quick Tip: When solving distance problems, always check your work by confirming the distance formula is applied correctly. A common mistake is forgetting to square the differences before adding them!

When given one point and the distance, you can find possible locations for the second point. This creates multiple solutions, as shown in problem #2 where x could equal either 5+3√2 or 5-3√2.

Applications with Circles

Circles in coordinate geometry bring these formulas to life. When you have a circle's diameter, you can easily find both its center and radius.

For example, if we have a diameter with endpoints B(3,-8) and C(6,-4), the center of the circle is at the midpoint of this diameter. Using our midpoint formula: (3+6)/2, (-8+-4)/2) = (9/2, -6) or (4.5, -6).

The radius equals half the diameter's length. We first calculate the distance between points B and C: d = √ $(3-6)² + (-8-(-4))²$ = √ $9+16$ = 5. Therefore, the radius is 5/2 or 2.5 units.

This relationship between diameter, center, and radius is fundamental for many geometry problems involving circles. You can use these coordinates to graph the circle or solve more complex problems.

Remember: A diameter always passes through the center of a circle, which is why finding the midpoint of the diameter gives us the circle's center!

Visualizing these points and circles on a coordinate plane helps solidify your understanding. Try sketching examples to see how the formulas translate to actual geometric shapes.

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Brad T

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Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

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Sudenaz Ocak

Android user

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Aubrey

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Elisha

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Paul T

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Subjects

Algebra 2

Modeling

Manipulating Formulas

Algebra 2

•

Dec 8, 2025

•

2 pages

Understanding Distance and Midpoint

Chelsea

@zuncho

Ready to master coordinate geometry? In this guide, we'll explore how to find the distance between points and calculate midpoints on a coordinate plane - key skills that help you solve real-world and theoretical math problems.

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Distance and Midpoint Formulas

Quick Tip: When solving distance problems, always check your work by confirming the distance formula is applied correctly. A common mistake is forgetting to square the differences before adding them!

When given one point and the distance, you can find possible locations for the second point. This creates multiple solutions, as shown in problem #2 where x could equal either 5+3√2 or 5-3√2.

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

Applications with Circles

Circles in coordinate geometry bring these formulas to life. When you have a circle's diameter, you can easily find both its center and radius.

The radius equals half the diameter's length. We first calculate the distance between points B and C: d = √ $(3-6)² + (-8-(-4))²$ = √ $9+16$ = 5. Therefore, the radius is 5/2 or 2.5 units.

This relationship between diameter, center, and radius is fundamental for many geometry problems involving circles. You can use these coordinates to graph the circle or solve more complex problems.

Remember: A diameter always passes through the center of a circle, which is why finding the midpoint of the diameter gives us the circle's center!

Visualizing these points and circles on a coordinate plane helps solidify your understanding. Try sketching examples to see how the formulas translate to actual geometric shapes.