Proportions

Key Concepts

• Properties of proportion
• Properties used in geometric polygons
• Similarity statements in geometric polygons

Topic/Sub-topic

Understanding of proportion properties 

If a−b / a+b = c−d / c+d, then prove that ab= c/d

Using the fourth property:  

(a−b+a+b)/(a+b) = (c−d+c+d) / (c+d)

2a / (a+b) = 2c / (c+d)

by cross product property 

2a(c + d) = 2c (a + b) 

2ac + 2ad = 2ac +2bc 

2ad = 2bc 

ad = bc 

By cross product property 

a/b = c/d

Use of proportions properties in geometric polygon 

• In the given figure, AB:BC= 5:8. Use the property of proportion to find AC:AB. 

Given, AB / BC = 5/8

AB+BC/AB = 5+8/5

(Using property 4 of proportions)  

AC/AB = 13/5

or AC/AB = 13/5   

In the given figure, AB/BC = FE/ED

Find AB. 

BC = 6, FE = 7, ED = 4 

AB/6 = 7/4

Using cross product property 

AB × 4 = 6 × 7 

AB = 6 × 7/4

  =21/2

 = 10.5 

Understanding proportion on geometric polygons 

Scale Drawing 

A map cannot be of the same size as the area it represents. So, it is scaled down to the size of a building or an area (map) in order to represent it on a paper or a plane. We use ratio for scale drawing. 

Scale drawing = drawing length: actual length 

In maps, map scale = map distance: actual distance 

Scale 1: 100000 means the real measure is 100000 times the length of the distance on a map. 

• A particular map shows a scale of 1:5000. What is the actual distance if the map distance is 10 cm? 

Scale = map distance: actual distance 

1:5000 = 10/actual distance 

1/5000 = 10/actual distance

actual distance = 10 × 5000 = 50000 cm  

= 50000/100 = 500m

Similar Polygons 

Two polygons are similar polygons if corresponding angles are congruent and corresponding side lengths are proportional. 

If two polygons are similar, then the ratio of the lengths of two corresponding sides is called the scale factor. 

• Triangle PQR~ABC. 

1. Write the congruent angles and corresponding sides. 
2. What is the scale factor? 
3. Why is it similar? 

Solution:

1. <P = <A, <R = <B, <Q = <C 

Corresponding sides =PR and AB, RQ and BC, PQ and AC 

2. PR/AB = 15/5 = 3/1, QR/BC = 18/6 = 3/1, PQ/AB = 3/1    

Scale factor= 3/1 

3. < P =< A, < R= < B, < Q= < C and PR/AB = QR/BC = PQ/AB = 3/1

Corresponding angles are congruent and corresponding sides are proportional, then triangles PQR and triangle ABC are similar. 

• In the diagram, ABC ~ DEF, the scale factor of ABC to DEF is 3:5. Find AC. 

Corresponding sides = AB and DE, BC and EF, AC and DF 

AB/DE = BC/EF = AC/DF = 3/5

AC/DF = 3/5

AC/12 = 3/5

5AC = 12 × 3 (cross product property) 

AC = 36/5

= 7.2 

Similarity statements in Polygons 

In the diagram, triangle LMN ~ triangle QRS. 

1. Find the scale factor of triangle LMN to triangle QRS. Then find the values of x and y. 
2. Find the perimeters of triangle LMN and triangle QRS. 
3. Find the areas of triangle LMN and triangle QRS. 
4. Compare the ratio of the perimeters to the ratio of the areas of triangle LMN to triangle QRS. What do you notice? 

Solution: 

1. LM/QR = MN/RS = QS/LN = 6/18 = 1/3                                                          

Scale factor =1/3

8/x = 6/18

x = 24 

since

6/18 = 4/y

y = 4 × 3 = 12 

2. Perimeter of triangle QRS = 6 + 6 + 8 = 20, perimeter of triangle RQS = 18 + 18 + 24 = 60 
3. Area of triangle QRS = 12 × 24 × 12 = 144, area of triangle LMN = 12 × 8 × 4 = 16 
4. Ratio of perimeters = perimeter of triangle LMN/perimeter of triangle QRS = 20/60 = 1/3 
5. Ratio of Areas = area of triangle LMN/area of triangle QRS = 16/144 = 1/9 

Ratio of the perimeter is equal to the scale factor of the corresponding sides of similar triangles, and the ratio of area is equal to the square of scale factor. 

Exercise

• If 2x + 3y : 3x + 5y = 18 : 29 find x : y
• Two numbers are in the ratio 3:5. If 8 is added to each number, the ratio becomes 2:3. Find the numbers
• If , Prove that .
• The measure of angles of a quadrilateral are in the ratio 1:2:3:4. Find the angles
• Two cities are 108 miles from each other. The cities are 4 inches apart on a map. Find the scale of the map.
• Given polygons are similar write corresponding angles and sides of the polygon.

• Find the scale factor of the following similar polygons

• Find the value of x for the given similar polygons

• A model satellite has a scale of 1 cm : 2 m. If the real satellite is 18 m wide, then how wide is the model satellite?
• Find the value of x

Concept Map

What we have learned

• Properties of proportion
• Properties used in geometric polygons
• Similarity statements in geometric polygons