* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download 9-4 the aaa similarity postulate
Survey
Document related concepts
Golden ratio wikipedia , lookup
History of geometry wikipedia , lookup
Line (geometry) wikipedia , lookup
Steinitz's theorem wikipedia , lookup
Multilateration wikipedia , lookup
Euler angles wikipedia , lookup
Atiyah–Singer index theorem wikipedia , lookup
Trigonometric functions wikipedia , lookup
Rational trigonometry wikipedia , lookup
Riemann–Roch theorem wikipedia , lookup
Noether's theorem wikipedia , lookup
Integer triangle wikipedia , lookup
Brouwer fixed-point theorem wikipedia , lookup
History of trigonometry wikipedia , lookup
Transcript
SIMILARITY Anggota: Desinta Yosopranata Devi Iriyani Elok Nur Afiyati M. Syafaur Rahmat Solekhah (4101414008) (4101414026) (4101414020) (4111411009) (4101410036) 4 cm 8 cm 5 cm 10 cm Both of these photographs are of the same subject, but one is larger than the other. Both pictures have the 'same shape'. If we compare the ratio of width to length of each picture, we see that the ratios are equal. This equation is called a proportion 4 8 because it is made up of two equal ratios 5 10 4 5 and 8 10 A proportion is an equality between two ratios. The a c ratios b and d are proportional if a c b 0, d 0 b = d . Example 1 In a proportion cross products are equal. 4 2 or 4x4 2x8 8 4 a c if b d then a x d = b x c Theorem 9-1 Example 2 In a proportion, I can be added to both sides. 9 12 9 3 12 4 9 3 12 4 If 3 4 then 3 3 4 4 or 3 4 ab cd a c if b d then b d Theorem 9-2 Example 3 In a proportion, I can be subtracted from both sides. 9 12 9 3 12 4 9 3 12 4 If 3 4 , then 3 3 4 4 or 3 4 a b cd a c If , then b d Theorem 9-3 b d Example If a c b d 4 ,then 9 12 3 4 9 3 12 4 Theorem 9-4 If a c b d ,then a b c d Example 5 9 12 If 9 x 4 = 3 x 12, then 3 4 If a x d = b x c, then Theorem 9-5 a c b d MN BC AM AN 1, 1 MB NC AM AN MB NC XY DE FX 1 FY 1 , XD 2 YE 2 FX FY XD YE Theorem 9-6 Side-Splitting Theorem. If a line is parallel to one side of triangle and intersect the other two sides, then it divides the two sides proportionally. PROOF Given: XY BC XY berpotongan dengan garis yang memuat CA , XY berpotongan dengan garis yang memuat AB Prove : AX AY XB YC Answers : A Y X C B Statements 1. XY BC Reasons 1. Given 2. XY Intersecting lines containing CA 2. Given 3. XY Intersecting lines containing AB 3. Given 4. X B ,X B 4. 5. A A 5. Berimpit 6. ABC XYA 6. AAA similarity postulate 7. AX AY XB YC 7. Restatement 6 XY BC Example 1 In this figure MN By theorem 9-6, hence or . Find NE! CM CN MD NE , DE 3 4 5 x 3x 20 x 20 3 Example 2 Find FJ in the figure shown if GF KJ. GK FJ By theorem 9-6, KH JH , 2 x 3 6 x 3x 12 2x or 5x 12 x 12 5 Theorem 9-7 If a line intersect two sides of a triangle and divides them proportionally, then the line is parallel to the third side. BUKTI Given :XY intersecting lines containing AC XY intersecting lines containing AB AX AY XB YC Prove : XY BC A Y X C B Statements Reasons 1. XY Intersecting lines containing AC 1. Given 2. Intersecting lines XY containing AB 2. Given AX AY 3. XB YC 3. Given 4. A A 4. Berimpit 5. Y C , X B 6. ABC 7. XY BC XYA 5. AX AY XB YC 6. AAA similarity postulate 7. Teorema kesejajaran garis yaitu Y C (sehadap) ABCDE A is similar to A’ B’ C’ E’. E E' A' B' B D' C' D C A A' , B B' , C C ' D D' , E E ' AB BC CD DE AE 2 A' B' B' C ' C ' D' D' E ' A' E ' WXYZ is similar to W’X’Y’Z’ Z Z' W Y X W' W W ' , X X ' , Y Y ' , Z Z ' WX XY YZ WZ 1 W ' X ' X 'Y ' Y ' Z ' W ' Z ' 2 X' Y' Notice that all corresponding angles are congruent and that the ratios of corresponding sides are equal. The symbol “ “ means “is similar to.” ABCD A’B’C’D’ means that ABCD is similar to A’B’C’D’. C B D D' A' C' B' A Two polygons are similar if there is a correspondence between vertices such that corresponding angles are congruent and corresponding sides are proportional. Example1 If we are given that ABCD A’B’C’D’, then we can conclude: 1. A A' , B B' , C C ' , D D' AB BC CD AD 2. A' B' B' C ' C ' D' A' D' D' B A C C' D B' A' Example 2 If we are given that 1. A A' , B B' , C C ' , D D' and 2. AB BC CD AD A' B' B' C ' C ' D' A' D' than we can conclude that ABCD A’B’C’D’. If you have a scaled map, then a figure on the map is similar to the figure it represents. Given a scaled map of locations A,B, and C, then . If A’C’=36 mm, A’B’=24mm, and AB=32 m, find AC. AB AC 32 AC Solution: or A' B' Therefore AC = C A 32m B A' C ' 24 36 32 36 48 24 In ΔABC and ΔDEF, <A <C <F. C <D, <B <E, and F D B A Observe that AB BC CA DE EF FD E It appears that whenever all three angles of one triangle are congruent to all three angles of another triangle, then the ratios of corresponding sides are also equal. We accept this as a postulate AAA Similary Postulate If three angles of one triangle are congruent to three angles of another triangles, then the triangles are similar. Theorem 9-8 AA Similarity Theorem. If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar Proof Given : ΔABC and ΔDEF with ˂A <D, <B <E Prove : ΔABC is similar to ΔDEF. C F D A B E Statements Reasons 1. A D 1. GIVEN 2. B E 2. GIVEN C F 3. Theorem 6-4 (The sum of the measure of the angles of a triangle is 180) 3. 4. 4. . AAA Similarity Theorem 9-9 Two right triangles are similar if an acute angle of one triangle is congruent to an acute angle of the other triangle Proof Given: B E, C F , A D Prove : ΔABC ΔDEF. A D B C F E Statements Reason B E 1. Given 2. C F 2. Given 1. 3. 3. Theorem 6-4 (The sum of the measure of the angles of a triangle is 180 ) A D 4. ABC DEF 4. AAA Similarity