1. Module 6

1.9. Page 4

Mathematics 10-3 Module 6 Lesson 2

Module 6: Triangles and Other Polygons

 

Bringing Ideas Together

 

In the Explore section you investigated another method of sketching similar polygons. You also reviewed the relationships among angles and among sides of similar polygons.

 

For two similar polygons, the angles are equal in measure, and the sides are proportional—that is, the ratios of corresponding sides are equal.

 

Self-Check

 

SC 5. How does the side proportion change for a similar polygon that is larger or smaller in size? Look at the similar polygons below.

 

This illustration shows three equilateral triangles in a row. The left triangle has 8-millimetre sides, the middle triangle has 4-millimetre sides, and the right-most triangle has 2-millimetre sides.

 

SC 6. The side proportion, or scale factor, moving between these triangles is shown. What do you notice about scale factor when you move from a large polygon to a small polygon? What does the scale factor look like as you move from a small polygon to a large polygon?

 

Compare your answers.

 

The following example determines if two figures are similar by comparing angle measurements and finding side ratios and scale factor.

 

Example 1

 

Are square floor tiles measuring 9 in on a side similar to square floor tiles measuring 1 ft on a side?

 

This illustration shows two coloured squares. The yellow square measures 9 inches on a side. The green square measures 1 foot on a side.

 

Solution

 

The corresponding angles are all right angles. So, the corresponding angles are equal in measure (congruent). A ratio is a comparison of two numbers. Make sure the units of all measurements are the same. If they are not the same, you will have to convert to a common unit. You have to use the same unit in both parts of the ratio.

 

Remember 1 ft = 12 in. So, the ratio of each pair of corresponding sides

 

Reduced to lowest terms, this ratio

 

Because corresponding angles are equal in measure and the sides are proportional, the squares are similar polygons.

 

Note that the scale factor describing the change from a 9-in square into a 12-in (1-ft) square is

 

 

Regular Polygons

 

regular polygon: a polygon with all its angles equal in measure and all its sides equal in measure

Did you know that some shapes are always similar to each other? All squares are similar. Squares are examples of regular polygons.

 

Equilateral triangles are another example of regular polygons. Each angle of an equilateral triangle is 60°, because 3 × 60° = 180°.

 

This illustration shows two equilateral triangles of different sizes.

 

Can you think of another example of a shape that is always similar?

 

Self-Check

 

SC 7. By what scale factor would you have to multiply the sides of an equilateral to obtain

 

This illustration shows two equilateral triangles, ABC and A'B'C'. The sides of triangle ABC measure 10 centimetres, the sides of triangle A'B'C' measure 5 centimetres.

 

SC 8. The kitchen in Jasper’s home is rectangular in shape and is 12 ft by 9 ft. Jasper’s bedroom is also rectangular and is 8 ft by 10 ft. Are the two rooms similar polygons?

 

Compare your answers.

 

Similarity

 

In the next example, you will investigate whether the given pentagons are similar.

 

Example 2

 

Determine the ratios of the pairs of sides of the pentagons ABCDE and Are the pentagons similar? Why or why not?

 

This illustration shows two pentagons. The pentagon on the left has a base of 8 centimetres, two vertical sides of 4 centimetres, two sides at the top of 5 centimetres and a top interior angle of 106 degrees. The pentagon on the right has a base of 4 centimetres, two vertical sides of 2 centimetres, two sides at the top of 2.5 centimetres and a top interior angle of 254 degrees.

 

Solution

 

Find the ratios of the sides.

 

 

 

The ratios of the sides are equal, but the figures are not similar. The corresponding angles are not equal, since 106° is not equal to 254°.

 

In the next examples you will find the missing sides of two similar figures by solving proportions.

 

Example 3

 

In the diagram the measurements of side lengths are shown in units. For example, has a measure of 2 units.

 

This illustration shows triangles ABC and ADE. DE is a segment that joins point D on side AB with point E on side AC. AD has a measure of 2 units. DB has a measure of 3 units. AE has a measure of 4 units. EF has a measure of 6 units.

 

Is Give reasons for your answer to this question.

 

View “Example 3 Solution: Similar Triangles.”

 

Example 4

 

Two similar triangular city lots are pictured below. Find the missing measures x and y.

 

This illustration shows two similar triangles. The triangle on the left has sides measuring 60 metres, 50 metres, and 40 metres. The angle opposite the 60-metre side is marked with a single arc. The angle opposite the 50-metre side is marked with a double arc. The angle opposite the 40-metre side is marked with a triple arc. The triangle on the right has sides measuring 30 metres, x, and y. The angle opposite x is marked with a single arc. The angle opposite y is marked by a double arc, and the angle opposite the 30-metre side is marked with a triple arc.

 

View “Example 4 Solution: Finding Lengths in Similar Triangles.”

 

Example 5

 

The outlines of two kites are similar polygons. Determine x.

 

This illustration shows two kites that are similar quadrilaterals. Each kite is made of two isosceles triangles. The quadrilateral on the left has two sides of 1 metre and two shorter sides of measure x. The quadrilateral on the right has two sides of 60 centimetres each and two sides of 36 centimetres each.

 

Solution

 

Did You Know?

 

This image shows the outline of a diamond-shaped kite with a tail made of triangles.

 

The Chinese were making kites from silk and bamboo 2800 years ago!

Remember that 1 m = 100 cm. Set up the proportion.

 

 

 

So, x = 60 cm.

 

Now it’s your turn.

 

Self-Check

 

SC 9. Dorothy is preparing to sew an outfit for her toddler. She wants to adapt the following pattern.

 

This illustration shows a pattern for a toddler’s dress.

 

To fit her child, Dorothy plans to reduce the dimensions of the pattern by 10%. What is the scale factor? What will the dimensions of this section be if Dorothy decides to round each measurement to the nearest quarter inch?

 

This illustration shows one piece of the toddler’s dress pattern. This quadrilateral is 14.5 inches long. The top measures 10.5 inches, and the bottom measures 17.5 inches.

 

SC 10. Shawn was writing a report about the World Cup held in South Africa in 2010. Shawn had to decide between two sizes of the South African flag for his report.

 

The length of the smaller flag is missing. Calculate the length to one decimal place.

 

These photographs show two different sizes of the South African flag. One measures 6 centimetres by 12 centimetres. The other measures 5 centimetres by x.

© Gino Santa Maria/1215500/Fotolia

 

SC 11. Maxine is interested in model aircraft. She wants to build a scale model of the Silver Dart, the first powered airplane to fly in Canada.

 

The photo shows the Silver Dart in flight.

<http://www.airforce.forces.gc.ca/v2/equip/hst/silverdart-eng.asp>; National Defence. Reproduced with the permission of the Minister of Public Works and Government Services, 2010



If the wingspan of the Silver Dart is 15 m, what will the wingspan of Maxine’s model be? Round to two decimal places.

 

Did You Know?

 

The original Silver Dart was made of steel tubing, bamboo, wire, and wood.



SC 12. The following polygons are similar. Find the missing measures (x, y, and z). Round to one decimal point.

 

This illustration shows two similar quadrilaterals. The quadrilateral on the left has sides measuring 4 centimetres, 6 centimetres, x, and y. The quadrilateral on the right has corresponding sides measuring 3 centimetres, z, 5 centimetres, and 2.5 centimetres.

 

Compare your answers.

 

Mastering Concepts

 

Answer this question. When you are finished, check your answer.

 

One rectangle that often appears in art and architecture is the golden rectangle. The proportions of the golden rectangle are considered pleasing to the eye. The length of the golden rectangle is approximately 1.618 times as long as it is wide. So, if the width is 1 m, the length would be 1.618 m.

 

This illustration shows a rectangle 1 metre tall and 1.618 metres wide.

 

One of the interesting properties of the golden rectangle is that if you draw a square inside with the width as one side, the rectangle remaining would also be a golden rectangle.

 

This illustration shows a rectangle 1 metre tall by 1.618 metres wide. A portion on the right side is coloured yellow. The width of this yellow rectangle is x. The uncoloured part of the large rectangle has a width of 1 metre.

 

Assuming the large golden rectangle is similar to the smaller golden rectangle on the right, use a proportion to find x, correct to 3 decimal places. Does this value of x make sense in the diagram? Why?

 

Compare your answer.

A regular pentagon is another example of polygons that are similar to each other. Each angle in a regular pentagon is 108°.

 

 

Like squares, all equilateral triangles are similar; all regular pentagons are similar, and so on. You may also have thought of all regular octagons, hexagons, heptagons, and so on.