## 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.

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

**Solution**:

- <P = <A, <R = <B, <Q = <C

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

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

Scale factor= 3/1

- < 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.

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

**Solution**:

- 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

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

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