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Introducing the Distance Formula
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Today, we are going to learn about the distance formula. This formula helps us calculate the distance between two points in the Cartesian plane. Can anyone tell me what that formula is?
Is it d equals the square root of the sum of differences?
Close! The formula is actually: $$d = \sqrt{(๐ฅ_2 - ๐ฅ_1)^2 + (๐ฆ_2 - ๐ฆ_1)^2}$$. Here, (๐ฅ_1, ๐ฆ_1) and (๐ฅ_2, ๐ฆ_2) are the coordinates of the two points. This is derived from the Pythagorean theorem.
Why is it derived from the Pythagorean theorem?
That's an excellent question! The distance between two points can be thought of as the hypotenuse of a right-angled triangle formed by the differences in their x and y coordinates.
Calculating an Example
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Letโs calculate the distance between points A(2, 3) and B(6, 7). Who can give me the step-by-step process using the distance formula?
First, we subtract the x-coordinates: 6 minus 2 equals 4.
Then we subtract the y-coordinates: 7 minus 3 equals 4 as well.
Exactly! Now, what do we do next?
We square the results, so 4 squared is 16 for both.
Great! Whatโs the next step?
We add those squares together: 16 plus 16 equals 32.
Correct! Finally, how do we find the distance?
We take the square root of 32, which is 4โ2!
Fantastic! You've just calculated the distance using the formula.
Introduction & Overview
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Quick Overview
Standard
In this section, we explore how to calculate the distance between two points on a Cartesian plane using the distance formula. We also examine a practical example to reinforce understanding.
Detailed
Detailed Summary
The distance between two points in a Cartesian coordinate system is calculated using the distance formula, which is derived from the Pythagorean theorem. Given two points, A (๐ฅ_1,๐ฆ_1) and B(๐ฅ_2,๐ฆ_2), the formula to find the distance (d) is:
$$
d = \sqrt{(๐ฅ_2 - ๐ฅ_1)^2 + (๐ฆ_2 - ๐ฆ_1)^2}
$$
This formula allows us to determine how far apart two points are on a two-dimensional plane. For instance, if we want to find the distance between points A(2, 3) and B(6, 7), substituting these values into the formula provides us with a result of 4โ2. Understanding this concept is essential as it forms the basis for further geometric analysis, such as finding midpoints and analyzing the slopes and equations of lines.
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Distance Formula
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The distance ๐ between two points ๐ด(๐ฅโ,๐ฆโ) and ๐ต(๐ฅโ,๐ฆโ) is given by:
๐ = โ((๐ฅโ โ ๐ฅโ)ยฒ + (๐ฆโ โ ๐ฆโ)ยฒ)
Detailed Explanation
The distance formula calculates the straight-line distance between two points on a Cartesian coordinate plane. The formula is derived from the Pythagorean theorem, which relates the lengths of the sides of a right triangle. In this case, if you imagine a right triangle formed by the two points ๐ด and ๐ต, the difference in their x-coordinates (๐ฅโ โ ๐ฅโ) gives you the length of one side, and the difference in their y-coordinates (๐ฆโ โ ๐ฆโ) gives you the length of the other side. By squaring these lengths and adding them together, you can find the square of the hypotenuse (the distance between the points). Finally, taking the square root gives you the actual distance.
Examples & Analogies
Think of finding a route between two locations on a map. The formula is like using a ruler to measure the shortest path between two points on that map, without having to account for any detours or turns you might take along a winding road.
Example Calculation
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๐ Example:
Find the distance between ๐ด(2,3) and ๐ต(6,7):
๐ = โ((6โ2)ยฒ + (7โ3)ยฒ) = โ(16 + 16) = โ32 = 4โ2
Detailed Explanation
In this example, we calculate the distance between points ๐ด(2,3) and ๐ต(6,7). First, we substitute the coordinates into the distance formula. The x-coordinates yield (6 - 2)ยฒ, which equals 4ยฒ = 16, and the y-coordinates yield (7 - 3)ยฒ, which also equals 4ยฒ = 16. Thus, we add these results together, giving us โ(16 + 16) = โ32. Since โ32 can be simplified to 4โ2, we conclude that the distance is approximately 5.66 units.
Examples & Analogies
Imagine measuring the distance between two points in a park. If point ๐ด is at coordinates (2,3) and point ๐ต is at (6,7), you're basically determining how far apart two picnic spots are, as if you were walking the straight line connecting both spots.
Definitions & Key Concepts
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Key Concepts
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Distance Formula: Determines the distance between two points in a 2D plane using the coordinates of those points.
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Pythagorean Theorem: A principle used to derive the distance formula, stating that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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To find the distance between points A(2, 3) and B(6, 7), use the distance formula to calculate a distance of 4โ2.
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The distance between points C(1, 1) and D(4, 5) calculates to โ((4-1)ยฒ + (5-1)ยฒ) = โ(9 + 16) = โ25 = 5.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
๐ต Rhymes Time
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To find the distance, hereโs the rule,
๐ Fascinating Stories
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Imagine two friends walking in a park, connecting points on a grid. To find the shortest path between them, they use a special recipe known as the distance formula, ensuring they wonโt walk too far.
๐ง Other Memory Gems
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To remember the distance formula: 'x2-x1, y2-y1, square it, sum it, take a root, and youโre done!'
๐ฏ Super Acronyms
D = Square the differences to calculate (D = SDยฒ), where SD is the side difference.
Flash Cards
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Glossary of Terms
Review the Definitions for terms.
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Term: Distance Formula
Definition:
A mathematical equation used to determine the distance between two points in a Cartesian plane.
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Term: Cartesian Plane
Definition:
A two-dimensional coordinate system defined by a horizontal axis (x-axis) and a vertical axis (y-axis).
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Term: Hypotenuse
Definition:
The longest side of a right triangle, opposite the right angle.