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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: . Here, (𝑥_1, 𝑦_1) and (𝑥_2, 𝑦_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.
Let’s 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! What’s 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 4√2!
Fantastic! You've just calculated the distance using the formula.
Determines the distance between two points in a 2D plane using the coordinates of those points.
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.
To find the distance, here’s the rule, Subtract the x’s and then the y’s too. Square those differences, then add them on. Take the square root, and the distance is drawn!
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 won’t walk too far.
To remember the distance formula: 'x2-x1, y2-y1, square it, sum it, take a root, and you’re done!'
D = Square the differences to calculate (D = SD²), where SD is the side difference.
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Distance Formula
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Pythagorean Theorem
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Hypotenuse
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A mathematical equation used to determine the distance between two points in a Cartesian plane.
A two-dimensional coordinate system defined by a horizontal axis (x-axis) and a vertical axis (y-axis).
The longest side of a right triangle, opposite the right angle.
This section introduces the distance formula used to calculate the distance between two points in a Cartesian coordinate plane.
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.
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 (𝑥_1,𝑦_1) and B(𝑥_2,𝑦_2), the formula to find the distance (d) is:
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 4√2. 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.