Similar Figures (Not exactly the same, but pretty close!)

Similar Figures Goal 1 Identify Similar Figures/Polygons Goal 2 Find the missing value Goal 3Describe a sequence that exhibits the similarity between two similar two‐dimensional figures

Congruent Figures In order to be congruent, two figures must be the same size and same shape.

What are similar Figures? Two figures are similar if corresponding (matching) angles are congruent and the lengths of corresponding sides are proportional.

Similar Figures Figures are said to be similar if : a)same figures but different in sizes. a)the corresponding angles are congruent a)their corresponding sides are proportional in lengths.

Similar Figures Similar figures must be the same shape, but their sizes may be different.

Sizes Although the size of the two shapes can be different, the sizes of the two shapes must differ by a factor

Sizes In this case, the factor is x

Similar polygons are polygons for which all corresponding angles are congruent and all corresponding sides are proportional. Example:

Angles and Sides in Similar Polygons Angles ∠A ≅ ∠ E ∠B ≅ ∠ F ∠C ≅ ∠ G Δ ABC ~ Δ EFG A BC E F G Sides AB ~ EF AC ~ EG BC ~ FG

Definition of Similar Polygons - Two polygons are SIMILAR if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional.

In the diagram, pentagon GHIJK is similar to (~) pentagon ABCDE, if all corresponding sides are proportional GHIJK ~ ABCDE

Similar triangles Two triangles are Similar if the only difference is size (and possibly the need to turn or flip one around).

Corresponding Sides In similar triangles, the sides facing the equal angles are always in the same ratio. For example

Determine the length of the unknown side ?

These triangles differ by a factor of ? 15 3= 5

Determine the length of the unknown side ?

These dodecagons differ by a factor of ? 2 x 6 = 12

Sometimes the factor between 2 figures is not obvious and some calculations are necessary ? =

To find this missing factor, divide 18 by ? =

18 divided by 12 = 1.5

The value of the missing factor is =

When changing the size of a figure, will the angles of the figure also change? ?? ? 70 40

Nope! Remember, the sum of all 3 angles in a triangle MUST add to 180 degrees. If the size of the angles were increased, the sum would exceed 180 degrees

70 40 We can verify this fact by placing the smaller triangle inside the larger triangle

70 40 The 40 degree angles are congruent.

Find the length of the missing side ?

This looks messy. Let’s translate the two triangles ?

Try this… ? 6

The common factor between these triangles is ? 6

So the length of the missing side is…?

That’s right! It’s ten!

Similarity is used to answer real life questions. Suppose that you wanted to find the height of this tree.

You can measure the length of the tree’s shadow. 10 feet

Then, measure the length of your shadow. 10 feet 2 feet

If you know how tall you are, then you can determine how tall the tree is by making a proportion. 10 feet 2 feet 6 ft

The tree must be 30 ft tall. 10 feet 2 feet 6 ft

Similar figures “work” just like equivalent fractions

These numerators and denominators differ by a factor of

Two equivalent fractions are called a proportion

Ratio and scale

What is scale? When we make a scale copy of an object, the original and the copy have the same proportions.

Ratio and models A model of a dinosaur has a scale of 1:12 This means that for every 1cm of the model the real life size is 12cm The head on the dinosaur model is 8 cm in length. How long is the head of the real dinosaur?

SCALE FACTOR The “real” measurement’s are multiplied or divided by the scale factor Bigger Life size ÷ Smaller Model/Plan X

Scale Factor The common ratio for pairs of corresponding sides of similar figures Example:

In figure, there are two similar triangles.  LMN and  PQR. This ratio is called the scale factor. Perimeter of  LMN = = 25 Perimeter of  PQR = = 18.75

Enlargements When you have a photograph enlarged, you make a similar photograph. X 3

Reductions A photograph can also be shrunk to produce a slide. x

Transformations on the Coordinate Plane Transformations are movements of geometric figures. The preimage is the position of the figure before the transformation, and the image is the position of the figure after the transformation.

Moving points Rules can be given to move coordinates on a plane to create a transformation Example: Using the following rule (x+1, y-2) what happens to the point (6, 8)? (6 + 1, 8 – 2) = (7, 6)

Making two points the same Triangle ABC has coordinates A(1, 2), B(3, 5), C( 6, 8) Triangle EFG has coordinates E(3, 5), F(5, 8), G(8, 11) Write a rule to change the vertices of triangle ABC to the vertices of triangle EFG.

Scale Factors of Perimeter and Area Perimeter: The ratio of the perimeters is the same as the ratio of the sides. Scale factor is the same too! Area: If the scale factor of the sides of two similar polygons is m/n, then the ratio of the areas is (m/n)^2. (You square each part of the ratio) –Example: Ratio of sides = ¾ Ratio of area = 9/16

Scale Drawing

Scale drawings Here is a plan of a garden greenhouse shed patio Scale 1: cm 7cm 9cm 6cm Find the actual length of the patio, shed & garden. Find the actual width of the greenhouse & garden 1.5cm

Some general rules All the “real” measurement’s are multiplied or divided by the scale factor Are you going from scale drawing or from life size? Bigger or smaller Bigger Life size ÷ Smaller Model/Plan X

Scale and maps Maps are made to scale. In each case, the scale represents the ratio of a distance on the map to the actual distance on the ground. … Scale is what makes map drawing possible. Every map has a scale printed on the front and you should always check this...

Scale and maps Below is a map of Oldham the scale is 2 : 200,000 You want to go to Oldham mumps. This measures 3 cms on the map. How far is the actual distance in Kilometers ? For every 2 cms on the map it is 200,000 cms in real life.

Test yourself A map of the Cotswolds. Scale 1inch = 4 miles You travel from Swindon to Tewkesbury. The distance is 42 miles, what is the measure on the map?

What did we just learn about similar polygons ? Same shape Different size Corresponding sideSize-change factor Equal angles Similar polygons

Example 1: Quadrangles ABCD and EFGH are similar. How long is side AD? How long is side GH? (1)What is the scale factor? (2)What is the corresponding side of AD ? (3)How long is side AD? (4)What is the corresponding side of GH? (5)How long is side GH? 12÷ 4= 3 & 18÷ 6=3 EH AD = 5 CD 7 x 3 = GH, GH = 21

Similar Figures Which of the following is not a characteristic of similar figures? A. Proportional sides B. Equal angle measurements C. Same shape D. Congruent side lengths

Similar Figures Are the figures below similar? Explain.

Similar Figures Are the figures below similar? Explain.

You can find the missing length of a side in a pair of similar figures, by using proportions 6 7 ΔRST ~ ΔUVW R S T U V W x ft. 6 ft. 35 ft. 7 ft. 35 = x x = 30 feet

Solve for x. x in. x in. 40 in. 30 in.. 40 = 12 x = 9 inches

Solve for x. Round to the nearest tenth. 4 x 12 in. 4 in. 20 in. x in.. 20 = 12 x = 6.7 inches

Solve for x m. 7 m. x 25 m. x = 14 x = 50 meters

Solve for x. Round to the nearest tenth. 15 x 17 in. x 35 in. 15 in. 17 = 35 x = 7.3 inches

Determine the missing sides of the triangle 39 in 24 in 33 in ? in 8 in ? in

Example 2 : Figure MORE is similar to Figure SALT. Select the right answer with the one of the given values below. (1) The length of segment TL is a. 6 cm b. 6.5 cm c. 7 cm d. 7.5 cmabcd (2) ER corresponds to this segment. a. TS b. TL c. AL d. SAabcd (3) EM corresponds to this segment. a. TS b. TL c. SA d. ALabcd (4) The length of segment MO. a. 6 cm b. 6.5 cm c. 7 cm d. 7.5 cmabcd M O RE T S A L

Determine the length of the missing side Two rectangles are similar. The first is 4 in. wide and 15 in. long. The second is 9 in. wide. Find the length of the second rectangle.

Determine the length of the missing side Two rectangles are similar. One is 5 cm by 12 cm. The longer side of the second rectangle is 8 cm greater than twice the shorter side. Find its length and width.

Test yourself Below are two measurements of a the real car. 3.7m 1.32m A model is made using a scale of 1:10 What is the length of the model? What is the height of the model