Lesson 1

Scale Drawings

  • Let’s make a scale drawing.

1.1: Is That the Same Hippo?

3 Hippos labeled Original, A, and B.

Diego took a picture of a hippo and then edited it. Which is the distorted image? How can you tell?

Is there anything about the pictures you could measure to test whether there’s been a distortion?

1.2: Sketching Stretching

A dilation with center \(O\) and positive scale factor \(r\) takes a point \(P\) along the ray \(OP\) to another point whose distance is \(r\) times farther away from \(O\) than \(P\) is. If \(r\) is less than 1 then the new point is really closer to \(O\), not farther away.

  1. Dilate \(H\) using \(C\) as the center and a scale factor of 3. \(H\) is 40 mm from \(C\).
    Segment C H is 40 millimeters long and horizontal. 
  2. Dilate \(K\) using \(O\) as the center and a scale factor of \(\frac{3}{4}\). \(K\) is 40 mm from \(O\).
    Segment K O is 40.

1.3: Mini Me

  1. Dilate the figure using center \(P\) and scale factor \(\frac12\).
    Figure with large circle A as body and small circle C as head. Segments H I and I J are left arm , segments E F and F G are right arm. 
     
  2. What do you notice? What do you wonder?


Horizontal line segment P Q, with point B. Perpendicular segment A B drawn. Lengths as follow: P B, 1. B Q, 2. A B, 1.
  1. Dilate segment \(AB\) using center \(P\) by scale factor \(\frac12 \). Label the result \(A'B'\).
  2. Dilate the segment \(AB\) using center \(Q\) by scale factor \(\frac12\).
  3. How does the length of \(A''B''\) compare to \(A'B\)? How would the length of \(A''B''\) change if \(Q\) was infinitely far away? Explain or show your answer.

Summary

A scale drawing of an object is a drawing in which all lengths in the drawing correspond to lengths in the object by the same scale. When we scale a figure we need to be sure to scale all of the parts equally or else the image will become distorted.

Creating a scaled copy involves multiplying the lengths in the original figure by a scale factor. The scale factor is the factor by which every length in a original figure is multiplied when you make a scaled copy. A scale factor greater than 1 enlarges an object while a scale factor less than 1 shrinks an object. What would a scale factor equal to 1 do?

For example, segment \(BC\) is a scaled copy of segment \(DE\) with a scale factor of \(\frac14\). So \(BC=\frac14DE\). If \(DE=6\), then \(BC=\frac64\) or 1.5.

Diagram showing line segment dilations with center A. 

To perform a dilation, we need a center of dilation, a scale factor, and something to dilate. A dilation with center \(A\) and positive scale factor \(k\) takes a point \(D\) along the ray \(AD\) to another point whose distance is \(k\) times farther away from \(A\) than \(D\) is.

Segment \(FG\) is a dilation of segment \(DE\) using center \(A\) and a scale factor of 3. So \(FA=3 \boldcdot DA\). If \(DA=15\), then \(FA=45\).

Glossary Entries

  • dilation

    A dilation with center \(P\) and positive scale factor \(k\) takes a point \(A\) along the ray \(PA\) to another point whose distance is \(k\) times farther away from \(P\) than \(A\) is.

    Triangle \(A'B'C'\) is the result of applying a dilation with center \(P\) and scale factor 3 to triangle \(ABC\).

  • scale factor

    The factor by which every length in an original figure is increased or decreased when you make a scaled copy. For example, if you draw a copy of a figure in which every length is magnified by 2, then you have a scaled copy with a scale factor of 2.