Lesson 2: Problem Solving - Drawing Similar Polygons

   Constructing Knowledge

A scale drawing is a representation of the original, either larger or smaller than the original. To create scale drawings, it is important to be able to draw similar polygons, either larger (enlargement) or smaller (reduction). The concept of scale is used in creating blueprints, mapping, and photography.

To create a similar (scale) drawing, you will need a ruler, a protractor and perhaps a calculator. You will need to measure the side lengths and the angles of the objects shown in the original drawing.

To draw a similar polygon, follow these steps:

  1. Label all vertices. The side lengths can be named according to the line segments connecting each pair of vertices.

  2. Measure each angle and side length. Record your findings.

  3. Multiply or divide each of the side lengths by the given scale factor.

  4. Draw one of the sides with its new measure.

  5. Measure the angle at one end of the new side, and make a small mark (the angles do not change in a scale drawing).

  6. Draw the next side, going through the small angle mark. This side length should correspond to the calculated length in step 3.

  7. Repeat steps 5 and 6 until all sides are drawn.

  8. Measure the last side (the one that connects to the first side drawn) and measure the angle between the first and last sides. If this side is not the correct length, or if the angle measure is incorrect, an error may have been made.

EXAMPLE 1

Draw a triangle similar to the one shown that is larger by a factor of 2.



Solution


Step 1: Label the vertices.



Step 2: Measure each angle and side length. Record your findings.

Angles original Side Length
∠A = 100° AB = 5.4 cm
∠B = 30° AC = 3.6 cm
∠C = 50° BC = 7 cm

Step 3: Multiply or divide each of the side lengths by the given scale factor.

The question states that the new triangle should be twice as big so multiply each side length by 2.

Angles original Side Length New Side lengths in Enlargement
∠A = 100° AB = 5.4 cm AB′= 10.8 cm
∠B = 30° AC = 3.6 cm AC′= 7.2 cm
∠C = 50° BC = 7 cm BC′= 14 cm

Step 4: Draw one of the sides with its new measure.

This example starts with side BC, but you can start with any of the sides.



Step 5: Measure the angle at one end of the new side, and make a small mark (the angles do not change in a scale drawing).

In this example, angle B (30°) is drawn, but angle C could be drawn as well. Remember that the angle measures do not change.



Step 6: Draw the next side, going through the small angle mark. This side length should correspond to the calculated length in step 3.

Line segment AB is drawn next, with a length of 10.8 cm



Step 7: Repeat steps 5 and 6 until all sides are drawn.

In this case, only one side is left so you can connect the sides. However, be sure to complete Step 8 to verify the accuracy of your scale drawing.



Step 8: Measure the last side and measure the angle between the first and last sides.

The last side measures 7.2 cm.
Angle C measures 50°.

Here are the original triangle and the enlargement:




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