Thinking about dilations | Transformations | Geometry | Khan Academy

Summary

This script demonstrates a geometric dilation of rectangle ABCP with the center at point P and a scale factor of 1 and 2/3. It explains how to find the image of the rectangle by calculating the new positions of points A, B, and C based on the scale factor. The script concludes with the lengths of side AB and its dilated image, showing a clear step-by-step process that results in AB being 6 units and its image 10 units long.

Takeaways

• 📐 The script discusses a geometric transformation involving a rectangle ABCP and a dilation centered at point P.
• 🔍 The dilation has a scale factor of 1 and 2/3, which means every point will be 1 and 2/3 times as far from P after the transformation.
• 📍 Point P remains at the same position since it is the center of the dilation.
• 📏 Point C's new position is calculated by multiplying its distance from P by the scale factor, resulting in a new y-coordinate of -7.
• 📐 The horizontal distance from P to point A is multiplied by the scale factor, moving it 5 units away from P's x-coordinate.
• 📏 Point B's new position is determined by applying the scale factor to both its horizontal and vertical distances from P, resulting in a new x-coordinate 5 units away and a y-coordinate 10 units below P's y-coordinate.
• 📏 The original length of side AB is 6 units, calculated from the difference between the x-coordinates of points A and B.
• 📏 The length of the image of side AB after the dilation is 10 units, calculated from the new positions of the points.
• 🔢 The scale factor of 1 and 2/3 is equivalent to multiplying by 5/3, which is used to calculate the new positions of the points.
• 📈 The process involves understanding how a dilation affects the position and distances of points in a geometric figure.
• 📚 The script serves as an educational resource for understanding geometric transformations such as dilations.

Q & A

What is the process described in the transcript?

The process described is a geometric transformation known as a dilation of a rectangle ABCP, with the center of dilation at point P and a scale factor of 1 and 2/3.

What is the scale factor used in the dilation?

The scale factor used in the dilation is 1 and 2/3, which is equivalent to 5/3 when expressed as an improper fraction.

How does the dilation affect the distance of points from the center P?

The dilation multiplies the distance of each point from the center P by the scale factor, making them 1 and 2/3 times as far away as they were originally.

What is the original position of point P in the dilation?

Point P is the center of dilation, so its position remains unchanged during the dilation process.

What is the original distance of point C from point P?

The original distance of point C from point P is 6 units, as it is 3 units below P's y-coordinate and 3 units to the left of P's x-coordinate.

How is the new position of point C calculated after the dilation?

The new position of point C is calculated by multiplying its original distance from P by the scale factor and adjusting the coordinates accordingly, resulting in point C being at (-3, -7).

What is the original length of side AB of the rectangle?

The original length of side AB is 6 units, as it is the horizontal distance between points A and B, which are 3 units away from P in opposite directions.

What is the length of the image of side AB after the dilation?

The length of the image of side AB after the dilation is 10 units, as it is calculated by multiplying the original length by the scale factor.

How is the new position of point A calculated after the dilation?

The new position of point A is calculated by multiplying its original horizontal distance from P by the scale factor, resulting in a new x-coordinate of 5 units to the right of P.

What is the significance of the dilation in geometric transformations?

Dilation is significant in geometric transformations as it allows for the enlargement or reduction of a shape while maintaining its proportions, with all points being scaled by the same factor from a central point.

What is the relationship between the original and the image of side AB in terms of length?

The length of the image of side AB is 1 and 2/3 times the length of the original side AB, demonstrating the effect of the dilation on the dimensions of the shape.

Outlines

00:00

📏 Dilation of Rectangle ABCP with Scale Factor 1 and 2/3

This paragraph describes a geometric transformation known as dilation applied to rectangle ABCP, with point P as the center of dilation and a scale factor of 1 and 2/3. The process involves calculating the new positions of the rectangle's points after dilation. Point P remains at the same location since it is the center of dilation. Point C, initially 6 units away from P, is repositioned to be 1 and 2/3 times further, resulting in a new distance of 10 units, placing the image of C at -7 in the y-direction. Point A's horizontal distance from P is increased by 1 and 2/3 times, moving it from 3 to 5 units away in the x-direction. The image of point B is calculated similarly, with its horizontal and vertical distances from P being scaled by the same factor. The original length of segment AB is 6 units, and after dilation, it becomes 10 units long, reflecting the scale factor applied.

Keywords

💡Dilation

Dilation in geometry refers to the transformation that enlarges or reduces a figure according to a specific scale factor. In the video's context, the dilation is centered at point P with a scale factor of 1 and 2/3, which means every point on the original figure will be magnified to 1 and 2/3 times its original distance from P. This concept is central to understanding the transformation of the rectangle ABCP.

💡Scale Factor

A scale factor is a number that indicates the ratio of the size of the image to the size of the original figure in a dilation. In the video, the scale factor of 1 and 2/3 (or 5/3 when expressed as an improper fraction) dictates the degree of enlargement of the rectangle's sides. It is crucial for determining the lengths of the sides of the dilated image.

💡Center of Dilation

The center of dilation is the point around which the dilation occurs. In the script, point P serves as the center, meaning all other points on the rectangle ABCP are moved away from or towards P by the scale factor. The center is fundamental to the dilation process as it remains unchanged while other points are transformed.

💡Rectangle

A rectangle is a quadrilateral with four right angles and opposite sides equal in length. In the video, rectangle ABCP is the original figure that undergoes dilation. The properties of a rectangle are important for understanding the relationships between the sides and how they change during the dilation.

💡Image of a Figure

The image of a figure in geometry is the result of a transformation, such as a dilation, applied to the original figure. In the video, the image of rectangle ABCP is the new figure obtained after the dilation with the center at P and the scale factor of 1 and 2/3.

💡X-coordinate and Y-coordinate

Coordinates in a Cartesian plane are used to locate points and are represented by ordered pairs (x-coordinate, y-coordinate). In the script, the x-coordinate and y-coordinate of points P, A, B, and C are used to calculate their positions after the dilation. Understanding coordinates is essential for visualizing and calculating the new positions of the points.

💡Horizontal Direction

The horizontal direction refers to the left-right orientation on a plane. In the video, the dilation affects the horizontal distances between points, as seen when point A moves from being 3 units to the right of P to 5 units after dilation.

💡Vertical Direction

The vertical direction refers to the up-down orientation on a plane. The script describes how the vertical distances are affected by the dilation, such as point C moving from being 6 units below P to 10 units below after the dilation.

💡Length of a Segment

The length of a segment is the measure of the distance between its endpoints. In the video, the length of segment AB is calculated both before and after the dilation to demonstrate the effect of the scale factor on the size of the figure. This concept is central to understanding the enlargement process.

💡Negative Coordinates

Negative coordinates indicate positions to the left of the y-axis or below the x-axis in a Cartesian plane. In the script, the negative coordinates of point C after dilation (-7) are used to illustrate the movement of points in the vertical direction during the dilation process.

💡Arithmetic Operations

Arithmetic operations such as addition and multiplication are used to calculate the new positions of the points after dilation. For example, the script uses multiplication to find 2/3 of the distances of the sides (e.g., 2/3 of 3 equals 2) and addition to determine the final coordinates (e.g., 3 plus 2 equals 5).

Highlights

The process involves a dilation of rectangle ABCP with center P and a scale factor of 1 and 2/3.

Every point after dilation will be 1 and 2/3 times as far from P as it originally was.

Point P remains at the same location after dilation since it is the center.

Point C's new position is calculated by multiplying its distance from P by the scale factor.

The calculation for point C's new y-coordinate is 3 - (6 * 1 and 2/3) = -7.

Point A's new horizontal position is found by applying the scale factor to its distance from P's x-coordinate.

Point A's new horizontal position is 5 units away from P's x-coordinate after dilation.

Completing the rectangle involves finding the new positions for points B and C.

Point B's new horizontal position is 5 units away from P's x-coordinate, after applying the scale factor.

Point B's new vertical position is 10 units below P's y-coordinate, calculated by the scale factor.

The original length of segment AB is determined to be 6 units.

The length of the image of segment AB after dilation is calculated to be 10 units.

The dilation process is demonstrated step by step, with clear calculations for each point's new position.

The dilation effect is visually represented by the change in the rectangle's dimensions.

The mathematical concept of dilation is applied to a geometric figure, providing a practical example of its use.

The transcript provides a clear explanation of how to calculate the image of a geometric figure under dilation.

Transcripts

The graph below contains the rectangle ABCP.

Draw the image of ABCP under a dilation whose center is at P

and a scale factor is 1 and 2/3.

What are the lengths of the side AB and its image?

So we're going to do a dilation centered at P.

So if we're centering a dilation at P

and its scale factor is 1 and 2/3,

that means once we perform the dilation,

every point is going to be 1 and 2/3 times as far away from P.

Well P is 0 away from P, so its image is still

going to be at P. So let's put that point right over there.

Now point C is going to be 1 and 2/3 times

as far as it is right now.

So let's see, right now it is 6 away.

It's at negative 3.

And P, its x-coordinate is the same, but in the y direction,

P is at 3.

C is at negative 3.

So it's 6 less.

We want to be 1 and 2/3 times as far away.

So what's 1 and 2/3 of 6?

Well, 2/3 of 6 is 4, so it's going to be 6 plus 4.

You're going to be 10 away.

So 3 minus 10, that gets us to negative 7.

So that gets us right over there.

Now point A, right now it is 3 more

in the horizontal direction than point P's x-coordinate.

So we want to go 1 and 2/3 as far.

So what is 1 and 2/3 times 3?

Well that's going to be 3 plus 2/3 of 3, which is another 2.

So that's going to be 5.

So we're going to get right over there.

Then we could complete the rectangle.

And notice point B is now 1 and 2/3 times

as far in the horizontal direction.

It was 3 away in the horizontal direction,

now it is 5 away from P's x-coordinate.

And in the vertical direction, in the y direction,

it was 6 below P's y-coordinate.

Now it is 1 and 2/3 times as far.

It is 10 below P's y-coordinate.

So then let's answer these questions.

The length of segment AB-- well, we already saw that.

That is, we're going from 3 to negative 3.

That is 6 units long.

And its image, well it's 1 and 2/3 as long.

We see it over here.

We're going from 3 to negative 7.

3 minus negative 7 is 10.

It is 10 units long.

We got it right.