Notes #36: Solving Ratios and Proportions and Similar Triangles (Sections 7.1 and 7.2) , 3 to 4, 3:4

Transcription

1 Name: Geometr Rules! Period: Chapter 7 Notes Notes #3: Solving Ratios and Proportions and Similar Triangles (Sections 7.1 and 7.) Ratio: a comparison of two quantities. 3, 3 to, 3: Proportion: two ratios that are equal to each other. 3 = 8 A. Solving Proportions Cross-multipl and set the products equal to each other Be sure to use FOIL when ou are multipling binomials together Solve for the variable Solve for : 1.) 3 =.) 7 = ) + 3 =.) = 3 B. Properties of Proportions

2 - - Complete: 3 5.) If =, then 7 =.) If, then 3 = = 7.) If a: = 5:3, then 3 a = 8.) If =, then = 9 9.) If =, then = 10.) If =, then = ) For the given figure, it is given that: KR =, KT = 10, KS = 8 K KR RT KS =. Solve for the missing lengths. SU RT = SU = R S KU = T C. Similar Polgons U Similar polgons have the same but not necessaril the same. Eample of similar triangles: B Y 10 3 X 5 Z A 8 C Their corresponding angles are Their corresponding sides are in a This ratio is called a and in this case is We show that the are similar with this statement:.) ABCDE A' B ' C ' D ' E ' C 9 D a) scale factor = b) m A' =, m D=

3 m C' = c) =, =, z = The figures are similar. Solve for the variables. (Hint: redraw the diagram as two figures) 13.) 18 z ) 1 9 D. Algebra Practice: Factor: p rs 1prs 1.) ) ) 3 3

4 - - Simplif: 18.) 3i ) 8i 3 0.) ( 3 5 ) Solve: 1.) 3 5 = 0.) + 10= 8 3.) = 7+ 10

5 Notes #37: Similar Triangles (Sections 7.3 and 7.5) Similar triangles have: corresponding angles sides that are in You can conclude that two triangles are similar if: : two pairs of corresponding angles are congruent : all three pairs of sides are in the same proportion : two pairs of sides are the same proportion and their included angles are congruent Are the triangles similar? If so, state the similarit and the postulate ou used.

6 Re-draw the triangles in matching positions Mark congruent angles arg Test sides for a constant proportion: small = medium = l e small medium l arg e Look for these patterns: AA~, SSS~, SAS~ ).) B 85 F 35 0 D O A 0 C E M N P Q 3.).) B F E 15 D F B 8 C 10 D A C A 5.) R.) Q X 5 S 9 Y Z State whether the figures are alwas, sometimes, or never similar: do the alwas, sometimes, or never have the eact same shape?

7 - 7-7.) two squares 8.) two congruent triangles 9.) two rectangles 10.) two rhombuses 11.) two pentagons.) two regular octagons Proportional Lengths (Section 7.5) A. Triangle Proportionalit A parallel slice cuts a triangle s sides proportionall ( Side-Splitter Theorem) j a c k a b = a, c = d, j = = k b b d a c a =, = j Eample: Solve for 0 33 B. Angle Bisector Proportionalit An angle bisector proportionall divides the opposite side

8 - 8 - w z z = w z = Eample: Solve for : w = C. Parallel Line Proportionalit Parallel lines proportionall divide their transversals c a b d a b = a c = Eample: Solve for : d c = 18-9

9 Solve for the variables: 1.).) ).) ).)

10 ) Write the equation of a line that contains the point ( -, 3) and has a slope 1 of. 15.) Write the equation of a line in standard form that is parallel to 3+ = 5 and contains the point ( 1, ). 1.) Write the equation of a line that contains ( -, -3) and (, -9) in standard form. 18.) Write the equation of a line that is perpendicular to = + 5 and contains the point ( -, 7)

11 Notes #38: Similarit in Right triangles ( 7.) Geometric Means and Similar Right Triangles A. Geometric Mean asks the question: what number, squared, equals the product of two given numbers? Find the geometric mean of the listed numbers: Use the given numbers in this equation: = ab Solve for 1.) 9 and 1.) and 3 3.) 5 and 15 B. Similar Right Triangles When an altitude of a right triangle is drawn to its hpotenuse, three similar right triangles are formed: z a b = a( a+ b) z = b( a+ b) = ( a)( b) Solve for the variables:

12 Re-draw the three triangles and label all sides Set up proportions to solve for the variables Look for was to use the Pthagorean theorem.) - - m n p 5 0 m = (5)(5) p = (0)(5) n = (5)(0) 5.) a b c

13 .) z

14 Notes #39 Section 8.1 Pthagorean Theorem In Words: In a triangle, the sum of the of the lengths of the is equal to the of the length of the. Pictures/Smbols: Eample: Find the missing side of the triangles below. 1.) ) 5 3 A Pthagorean Theorem. is a set of whole numbers a, b, and c, that satisf the Eamples: Do the lengths of the sides given form a Pthagorean triple? 3.) 8, 15, 17.) 7,, 5.) 0, 1, 9 Eamples: Find the value of. Leave our answer in simplest radical form..) 7.) ) 1

15 Determining Whether a Triangle is Right, Acute, or Obtuse Given Three Side Lengths: Right Acute Obtuse E 9: Sides have lengths 3,, and 5 E 10: Sides have lengths,, and 11 E 11: Sides have lengths 1, 7 and Eamples: The lengths of the sides of a triangle are given. Classif the triangle as acute, right, or obtuse..) 10, 15, 0 13.) 7,, 1.) 15, 0, 5 Eamples: Find the value of. Leave our answer in simplest radical form. 15.) 1.) Algebra Review: Solve using quadratic formula

16 Eamples: 17.) + 5= 0 18.) = ) 3 8= 0.) = + Notes #0: Chapter 7 Review

17 Simplif each ratio: 1.) a) BC:CD b) m<b:m<c B 8 C c) CD: Perimeter of ABCD 0 A D.) If =, =, z = find each ratio: a) to b) ( + z) to c) + 7z 3.) ab ab 5 7.) + for = 3, =, z = 1 z Write and equation and solve:.) The ratio of the angles of a triangle is 1:3:5. Find the angles. 5.) The ratio of the angles of a pentagon is : 8: 9: 11: 11. Find the angles.

18 Are the triangles similar? If so, write a similarit statement and the postulate ou used:.) C 7.) B A 10 E D 8.) 9.) E 9 N P D 15 F O Y M Q X Z Solve for the variables: 10.) = ) 3.)

19 ) 1.) Simplif: 15.) Similar Right Triangles: Solve for m, n, and p in reduced radical form. 1.) a.) Find the geometric mean of 5 and 10 m n p b.) Find the geometric mean of and Are the figures sometimes, alwas, or never similar? 17.) two rectangles 18.) two equilateral triangles 19.) two regular heagons STUDY GUIDE 7 Name:

20 - 0 - Show all our work! Date: Period: For #1-3, ABCD is a parallelogram. Simplif each ratio: 1.) BC:CD 1.) A B.) AD:(Perimeter of ABCD) 8.) 3.) m A: m B 0 D C 3.) For #, complete each statement:.) If a : 3 = 7 :, then a = 5.) If = 3 = 8, then.) If =, then 3 + =.) 5.).) For #7-10, solve for : ) = 8.) = ) = 8.) = 9.) 1 + = + 10.) 3 1 = ) = 10.) = For #11 1, state whether the two polgons are alwas, sometimes, or never similar. 11.) two right triangles 13.) two squares 15.) an isosceles triangle and a right triangle.) two scalene triangles 1.) two rectangles 1.) two regular heagons 11.).) 13.) 1.) 15.) 1.)

21 - 1 - For #17-0, refer to the diagram. 17.) Find m M ' 17.) 18.) Find the scale factor of MATH to M A T H A T 18.) 19.) Solve for. M 70 T' 18 A' H 19.) = H' M' 0.) = 0.) Solve for. MATH ~ M A T H For #1 3, complete the similarit statement and state wh the triangles are similar. If the triangles are not similar, circle not similar. (If ou are using SAS similarit or SSS similarit, be sure to check our side lengths for a common proportion) 1.) Q R X 9 S 1 8 Z Y.) N M O P Q 1.) ΔQRS Δ b OR not similar.) ΔMNO Δ b OR not similar

22 - - 3.) F.) A B 3.) ΔABC Δ b B A 9 C 8 D D C 1 30 E OR not similar.) ΔABC Δ b OR not similar For #5 8, solve for and (where and are positive): 5.).) 5.) = 1 7 = 9.) = 7.) 8.) 15 7.) = ) = Solve: 9.) + 1= 30.) + 0= 0 31.) 3 3 = 8 3.) 3 + 5= 9 9.) 30.) 31.) 3.)

23 For #33-3, find the geometric mean of the two numbers. 33.) 5 and 10 3.) and 0 33.) ) For #35-3, solve for,, and z. (Hint: use 3 similar, right triangles) 35.) 35.) = 5 z = z = 3.) 3.) = 1 z = z = For # 37-38, solve for. Leave the answer in simplified radical form 37) 38.) 37.) = 38.) = 9 For #39-0, the lengths of the sides of a triangle are given. Classif each triangle as acute, right, or obtuse. 39.) 0, 30, 0 0.) 1, 9, 0 39.) 0.)