7.4 - Summary
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Distance Formula
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Today, we'll start with the Distance Formula. Does anyone remember how we calculate the distance between two points in a coordinate plane?
Isn’t it something like the Pythagorean theorem?
Exactly! We use the Pythagorean theorem to find the distance. The formula is: \[ PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]. Can anyone explain what the variables represent?
x1 and y1 are the coordinates of the first point, and x2 and y2 are the coordinates of the second point.
Perfect! And remember, to simplify this, you could also express this in the squared form first, right?
Yes! It’s good to know the square version first, to avoid calculation mistakes.
Great insight! To recall, we are just applying the Pythagorean theorem where the distance is the hypotenuse.
Can you give an example to illustrate this?
Sure! For points A(2, 3) and B(5, 7), the distance would be \[ PQ = \sqrt{(5-2)^2 + (7-3)^2} = \sqrt{3^2 + 4^2} = 5. \] This is the formula in practice!
So, what do we conclude about the distance formula?
We can calculate the distance between any two points using their coordinates!
Exactly! Let’s summarize the distance formula before we move on.
From the Origin
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Now that we understand the general distance formula, let’s focus on finding the distance of a point P(x, y) from the origin (0, 0). What formula can we use here?
Is it just the same distance formula but with the origin coordinates?
Exactly! We simplify it down to \[ OP = \sqrt{x^2 + y^2} \]. Can someone think of practical implications?
This would help in graphing information in the first quadrant, right?
Perfect! Distance from the origin is often useful in physics and other fields where we analyze points related to a starting factor. Let’s break this down by computing an example. What about the point P(3, 4)?
Using the formula, it would be \[ OP = \sqrt{3^2 + 4^2} = 5. \]
Excellent! Does everyone see how this works?
Yes! Whenever we need to find the position of a point in relation to the origin, we can use this formula.
Right! Now let me give you another example to solve shortly. Let’s proceed to the next topic.
Section Formula
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Next, let’s tackle the section formula. This helps us find the coordinates of a point, P, that divides the line segment between two points A and B in a given ratio. Who can express the section formula for me?
It’s \[ P = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \], right?
Correct! And what do m and n represent here?
They represent the ratio in which the segment is divided.
Exactly! Let’s say from A(2, 3) to B(5, 7) and we want to divide it in the ratio 1:2. What would we compute?
Using the formula, we can compute the coordinates for point P.
Correct! Let's calculate that together, starting with the coordinates. Who can substitute the numbers for me?
I can! It will be \[ P = \left(\frac{1*5 + 2*2}{1+2}, \frac{1*7 + 2*3}{1+2}\right) = \left(\frac{9}{3}, \frac{13}{3}\right) = (3, \frac{13}{3}) \].
Spot on! This point gives us a clear landmark on the segment dividing it in a 1:2 ratio. Remember, this formula is essential in both geometry and real life for determining positions, such as where to place objects.
This will help us in more complex problems involving linear graphs!
Exactly! Let’s summarize the section formula before we move forward.
Midpoint Formula
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Finally, let’s discuss the midpoint formula, which is quite simple yet very useful. The midpoint M between points P(x1, y1) and Q(x2, y2) is defined as \[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \].
So it’s averaging both x and y coordinates?
Exactly! Let’s look at an example — for points A(1, 2) and B(3, 6) what would be the midpoint?
It will be \[ M = \left(\frac{1 + 3}{2}, \frac{2 + 6}{2}\right) = (2, 4). \]
Perfect! Finding midpoints can be particularly handy when constructing geometric shapes or in real-world applications like navigation.
That’s useful because we could find the center of a line segment which is crucial in graphics!
Great observation! Let's quickly recap the formulas we've covered, shall we?
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
The summary encapsulates key concepts in coordinate geometry, such as how to determine the distance between two points in a plane, the coordinate of a point that divides a segment in a given ratio, and how to find the midpoint between two points.
Detailed
In this section, we delve into essential formulas and concepts of coordinate geometry. We learned that the distance between two points P(x1, y1) and Q(x2, y2) in a Cartesian plane is given by the formula
Distance Formula:
\[ PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Additionally, we studied that the distance of a point P(x, y) from the origin (0, 0) can be expressed as
From the Origin:
\[ OP = \sqrt{x^2 + y^2} \]
We also explored the section formula, which finds the coordinates of point P dividing the segment joining A and B in the ratio m: n, given by
Section Formula:
\[ P = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]
Lastly, the midpoint of a line segment connecting points P and Q is calculated as
Midpoint Formula:
\[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
Understanding these formulas lays the foundation for more complex applications and problem-solving in coordinate geometry.
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Distance Formula Between Two Points
Chapter 1 of 4
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Chapter Content
The distance between P(x , y ) and Q(x , y ) is (x x )² + (y y )².
1 1 2 2 2 1 2 1
Detailed Explanation
This formula calculates the distance between two points P and Q on a coordinate plane. It involves taking the difference between the x-coordinates and the y-coordinates of the two points, squaring these differences, summing them, and then taking the square root to find the actual distance.
For example, if we have P(3, 4) and Q(7, 1), the distance formula will evaluate as follows:
- Calculate the differences:
- x-coordinate difference: 7 - 3 = 4
- y-coordinate difference: 1 - 4 = -3
- Square the differences:
- (4)² = 16 and (-3)² = 9
- Add the squares:
- 16 + 9 = 25
- Take the square root:
- √25 = 5
Thus, the distance is 5 units.
Examples & Analogies
Imagine you are standing at point P (3, 4) on a park map, and you want to walk to point Q (7, 1). By using the distance formula, you can easily calculate how far you have to walk to reach your friend at point Q. Think of it as finding the direct path between two locations rather than moving along the grid lines of the park.
Distance from the Origin
Chapter 2 of 4
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Chapter Content
The distance of a point P(x, y) from the origin is x² + y².
Detailed Explanation
This formula provides a way to find how far a point P is from the origin (0, 0) on a coordinate plane. To calculate this distance:
1. Square the x-coordinate of the point.
2. Square the y-coordinate of the point.
3. Sum these squares.
4. Take the square root. This gives us the actual distance from the origin.
For example, for the point P(3, 4):
1. (3)² = 9
2. (4)² = 16
3. 9 + 16 = 25
4. √25 = 5.
Thus, the distance from point P to the origin is 5 units.
Examples & Analogies
Picture being on a game board, where the center is a starting point (the origin). As you move to point P(3, 4), the distance from the center tells you how far you are from the beginning of the game. It’s like measuring how far you have advanced from your starting line.
Section Formula for Internal Division
Chapter 3 of 4
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Chapter Content
The coordinates of the point P(x, y) which divides the line segment joining the points A(x , y ) and B(x , y ) internally in the ratio m : m are
1 1 2 2 1 2
(mx₁ + mx₂, my₁ + my₂)/(m₁ + m₂).
Detailed Explanation
This formula helps us find the coordinates of a point P that divides the line segment joining two points A and B in a specific ratio. It combines the coordinates of both points A and B, weighted by the ratios provided.
For example, if A(2, 3) and B(4, 5) are to be divided in the ratio 1:2, we substitute:
- m₁ = 1, m₂ = 2, x₁ = 2, y₁ = 3, x₂ = 4, y₂ = 5.
- Thus,
x = (12 + 24)/(1 + 2) = (2 + 8)/3 = 10/3 = 3.33
y = (13 + 25)/(1 + 2) = (3 + 10)/3 = 13/3 = 4.33
So, the coordinates of P are approximately (3.33, 4.33).
Examples & Analogies
Think of two friends standing at points A and B along a path. If you want to know where someone would be if they walked from A towards B, covering one-third of the distance of the total path towards their friend, that person’s position can be determined using the section formula. It helps visualize and calculate their exact position along the path.
Mid-point Formula
Chapter 4 of 4
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Chapter Content
The mid-point of the line segment joining the points P(x , y ) and Q(x , y ) is
1 1 2 2
(x₁ + x₂, y₁ + y₂)/2.
Detailed Explanation
The mid-point formula calculates the exact center point between two given points on a coordinate plane. To find the mid-point:
1. Add the x-coordinates of both points.
2. Add the y-coordinates of both points.
3. Divide both sums by 2.
For example, the mid-point between P(2, 3) and Q(4, 5) is:
- x = (2 + 4)/2 = 6/2 = 3
- y = (3 + 5)/2 = 8/2 = 4
Thus, the mid-point is (3, 4).
Examples & Analogies
Imagine two friends, Alice and Bob, who live in two different homes on a straight road in the neighborhood. If they want to meet halfway at a park, using the mid-point formula helps them find that exact spot together on their graph of houses.
Key Concepts
-
Distance Formula: A method of calculating the distance between two points in a coordinate plane.
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Section Formula: Used to find a point that divides a line segment in a given ratio.
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Midpoint Formula: Calculates the average position of two points to find the midpoint.
Examples & Applications
Example 1: For points A(1, 2) and B(3, 4), the distance is calculated as PQ = √[(3-1)² + (4-2)²] = √[2² + 2²] = √8 = 2√2.
Example 2: To find the midpoint M of A(4, 6) and B(8, 10), M = [(4+8)/2, (6+10)/2] = (6, 8).
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
When finding lengths, remember this song, Use two points, it won’t take long. Just square each part, add and then root, That’s how you find the distance, it’s cute!
Stories
Once in the land of Geometry, a wise teacher explained that each point has its place, and to find distances between them, it becomes a magical adventure using formulas!
Memory Tools
To recall Distance Formula: D = square root of dx² + dy², where dx and dy are changes in x and y.
Acronyms
For the Section Formula, remember 'MAP' - M(x1, x2), A(m, n) = P(p1, p2) where P resembles the Position.
Flash Cards
Glossary
A formula used to determine the distance between two points in a coordinate plane.
A formula to find the coordinates of a point that divides a segment in a specific ratio.
A formula to calculate the midpoint of a segment between two points.
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