Successfully reported this slideshow.
Upcoming SlideShare
×

# Similar Triangles

41,667 views

Published on

Scale Factor, Similarity, and Triangles

• Full Name
Comment goes here.

Are you sure you want to Yes No
• This is a really great slide show on similar shapes, similar triangles

Are you sure you want to  Yes  No
• can I borrow this for my report only in geometry please

Are you sure you want to  Yes  No

### Similar Triangles

1. 1. Tourmaline Crystals – Image Source: http://www.mnh.si.edu
2. 2. When objects are the exact same shape, but they are different sizes, we say that they are “Similar” objects or figures. Images from Google Images
3. 3. When we double the Length and Width of our Photo we ENLARGE it using a SCALE FACTOR of 2. When we halve the Length and Width of our Photo we REDUCE it using a SCALE FACTOR of 1/2.
4. 4. If the SCALE FACTOR is Greater than 1 the resulting object is an ENLARGEMENT. If the SCALE FACTOR is Less than 1 the resulting object is a REDUCTION. Scale Factors less than 1 are expressed as Fractions: ¼, ½, or as decimals: 0.3 etc 3 4 6 8
5. 5. We calculate the SCALE FACTOR by comparing matching sides, using Ratios. We always Compare the New shape, with the Original shape (Big vs Small for these Photos). New 6 8 Old 3 4 3 4 6 8 = 2== For our Photo Enlargement we have: S.F. =
6. 6. When we compare the Ratios of the matching sides, we get “2” in both cases. We say the two photos are “Similar” : eg. Exact same shape, but different sizes. New 16 9 Old 4 3 4 3 8 6 NOT == For Photo Enlargement 2 we have: S.F. = 4 3 16 9 When the Ratios are not all the same, the objects are not similar to each other.
7. 7. We know for certain that they are the same shape, because they contain the same three angles. (This is the AAA rule for Similar Triangles). E A Similar Triangles are the exact Same Shape, but are Different Sizes. ABC ~ DEF (~ means similar to) 12cm 6cm 65o 65o 40o 40o C B FD 16cm 8cm10cm 5cm 75o 75o
8. 8. AC 12 DF 6 The Triangles are the same shape, because their three Angles are identical. (We do not have to calculate the third angle because we know it is 180 minus the other two angles in both cases). AB 10 DE 5 E A = 2= The Ratios of the matching sides are all the same value. (S.F. = 2) This proves the Triangles are Similar. ABC ~ DEF (~ means similar to) 12cm 6cm 65o 65o 40o 40o C B FD 16cm 8cm 10cm 5cm BC 16 EF 8 = 2= = 2=
9. 9. E A Similar Triangles are the exact Same Shape, but are Different Sizes. If two different triangles contain the exact same angles, but are different sizes, then they are similar. We call this the AAA Rule. ABC ~ DEF (~ means similar to) 65o 65o 40o 40o C B FD 75o 75o
10. 10. AC 12 DF 6 These two Triangles are SIMILAR, because their three Sides are all Proportional. (Eg. When we calculate for the matching sides, they all give the same Scale Factor. AB 10 DE 5 E A = 2= The Ratios of the matching sides are all the same value. (S.F. = 2) All Sides are Proportional to each other. Triangles are Similar by PPP Rule. ABC ~ DEF (~ means similar to) 12cm 6cm C B FD 16cm 8cm 10cm 5cm BC 16 EF 8 = 2= = 2=
11. 11. AC 12 DF 6 Two Triangles are similar if two of their matching sides are in Proportion, and the included angle between these sides is the same in both triangles. (This can be proven by drawing lots of triangles and measuring the three sides). E A The Ratios of the matching sides are all the same value. (S.F. = 2) They have the same 40 degree angle. Triangles are Similar by PAP or SAS. ABC ~ DEF (~ means similar to) 12cm 6cm 40o 40o C B FD 16cm 8cm BC 16 EF 8 = 2= = 2=
12. 12. E A If two different sized Right Triangles contain a 90 degree angle, and one matching side, as well as the Hypotentuse are in the same Proportion, then they are Similar by the RHS Rule. ABC ~ DEF (~ means similar to) C B F D 8mm 4mm 10mm 5mm
13. 13. Tourmaline Crystal cross sections contain Similar Triangles
14. 14. Tourmaline is found in Mozambique, and is a gem used to make spectacular jewellery such as these colorful cufflinks.
15. 15. Similar Triangles can also be used to great effect in Art and Craft, as seen in this colourful and creative patchwork quilt. Image Source: http://www.patchworkpatterns.co.uk
16. 16. Many Similar Triangles are present in this modern art piece. Image Source: http://www.trianglesoflight.org
17. 17. Many Similar Triangles are present in the structure of the Sydney Harbour Bridge to give it strength and stability. Image Source: Copyright 2012 Passy’s World of Mathematics
18. 18. A B C All we need to do is calculate the missing third angle C = R = 180 - (85 + 55) = 180 – 140 = 40o ABC ~ PQR (by AAA Rule) Use either AAA or PPP or PAP or RHS to prove these are Similar Triangles 5m 85o P Q R 15m 85o 55o 55o
19. 19. Now we need to check the Ratios of the matching sides C = F = 70o ABC ~ DEF (by PAP or SAS Rule) Use either AAA or PPP or PAP or RHS to prove these are Similar Triangles A B C 8 70o D E F 12 70o 3 4.5 DF 12 AC 8 = 1.5= EF 4.5 BC 3 = 1.5=
20. 20. A B C We know the third angle is 40o so ABC ~ PQR by AAA. Prove these are Similar Triangles and then find the value of “n” 5 85o P Q R 15 85o 55o 55o 10 n = ? PQ 15 AB 5 = 3= This Scale Factor of S.F. = 3 tells us that PQR is three times bigger than ABC. Using the Scale Factor of 3: n = 3 x 10 = 30
21. 21. A B C We know the third angle is 40o so ABC ~ PQR by AAA. Prove these are Similar Triangles and then find the value of “n” 5 85o P Q R 15 85o 55o 55o 10 n = ? PQ PR AB BC = Substitute n 15 10 5 = Cross Multiply n 15 10 5 = 5n = 150 Divide by 5 n = 30
22. 22. A B C We know the third angle is 40o so ABC ~ PQR by AAA. Prove these are Similar Triangles and then find the value of “k” 30 85o P Q R 15 85o 55o 55o 10k = ? PQ PR AB BC = Substitute 15 30 k 10 = Cross Multiply 15 30 k 10 = 30k = 150 Divide by 30 k = 5