Thinking about dilations | Transformations | Geometry | Khan Academy

Khan Academy
16 Jul 201502:39

Summary

TLDRThis 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.

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