Enlargements and Reductions

Chapter 7.1

Enlargement and Reduction

• An enlargement or reduction involves making it larger or smaller (respectively).

• The ' symbol is called the prime symbol. It is used to show the vertex on an image that corresponds with a vertex on the original diagram.

Scale Factor

• If the scale factor is a number greater than one, the shape was made larger.

• If the scale factor is a number less than one, the shape was made smaller.

• Scale factor = Image Length/Original Length

To find the scale factor:

Steps:

1) Pick one known length in the image diagram.

2) Find the corresponding known length in the original diagram.

3) Use the general formula to calculate the scale factor. Divide the length in the image diagram by the corresponding length in the original diagram.

Similar Triangles

Properties of Similar Triangles

• All corresponding angles are equal

• All corresponding sides are proportionately equal in length.

This means that similar triangles have the same shape, but are not necessarily the same size.

For two triangles to be equal we must prove:

That either:

Problem Solving with Similar Triangles

Steps

1) Indentify the corresponding sides in both triangles

2)Write a proportion that represents the ration of each pair of corresponding sides.

3)Solve for x by applying the cross products. Chose a ratio where you know both the numerator and denominator of the fraction, and the ration with the variable that you need.

4)Solve for the variable

Similar Polygons

• A polygon is any closed plane figure constructed by three or more line segments

• A quadrilateral is a 4-sided polygon such as a square, rectangle, trapezoid and parallelogram.

• Similar polygons share the same skills learnt with similar triangles.

Properties of Similar Polygons

• Matching pairs of angles that are found in the same place (corresponding angles) in both polygons are equal

• Matching pairs of sides that are found in the same place (corresponding sides) in both polygons are proportional

Example

Example – Draw a similar polygon, state the scale factor used.

• Remember, the polygons (and triangles) don’t need to be in the same rotation to be similar. They can be flipped or rotated and still be similar.

Are they similar? Justify your answer

If you are asked to find an unknown side of a polygon that is similar to another polygon you need to:

Set up a proportion that:

1) Shows the ratio of corresponding side lengths.

2) Solve the unknown side by applying cross products.

Review Summary

• Scale drawings are used when objects are too large or too small to be drawn on a sheet of paper

• There are two properties that pertain to similar polygons (which include similar triangles):

– Corresponding angles are equal in measure

– Corresponding sides are proportional in length