1

Ratio and Proportion Ratio – comparing of two quantities Written two ways: x : y x y A Proportion is an equation that states that two ratios are equal(cross Product). a = b c d ad = bc In a proportion, the product of the means is equal to the product of the extremes

bc = ad

Mean Proportion

If two means of a proportion are equal, either mean is called the mean proportional between the extremes of the proportion In which of the following may the ratios form a proportion:

1. 2. 3. 4. 5.

Use each set of numbers to form two proportions: 6. 40, 10, 1, 4 7. 4, 6, 18, 12 8. 2, 9, 6, 3 9. 28, 6, 24, 7

Determine which of the following is a true statement: 10. 5 : 10 = 10 : 20 11. 3 : 4 = 15 : 20 12. 12 : 18 = 36 : 72

2

Solve the proportion for x:

13. 14.

15.

16. 17.

18.

19. x : 60 = 6 : 10 20. 16 : x = 12 : 9

21. x : 10 = 65 : 5

Given; XZ = 4 and ZY = 6. State the numerical value of each ratio. 22. XZ : ZY

23. ZY : XZ

24. XZ : XY

25. XY : ZY

3

Find the mean proportion between: 26. 4 and 9

27. 2 and 32 28. 4 and 25 29. .27 and .03

30. 27 and 3

31. 4 and 16 32. 8 and 12 33. b and d

In triangle ABC, : or ΔADE ~ ΔABC

34. If AD = 5, DB = 15 and AE = 8, Find EC

35. If AD = 2, AE = 6 and EC = 18, find DB.

36. If DB = 6, AE = 12, and EC = 24, find AD.

37. If AB = 25, AD = 10, and AE = 8, find EC.

38. If AB = 10, DB = 2 and EC = 3, find AE.

39. If AD = 4, AE = 12 and EC = 36, find DB.

4

In ΔABC, CD : DB = DB : DA. 40. If CD = 4 and DB = 10, find AD.

41. If CD = 2 and DB = 8, find AD.

42. If CD = 3 and DA = 12, find DB.

43. If CD = 3 and DA = 48, find DB.

Midpoint Theorem If a line segment joins the midpoints of two sides of a triangle the segment is parallel to the third side and its length is one-half the length of the third side. Example: In ΔABC, D is the midpoint of AB and E is the midpoint of AC. If BC = 7x + 1, DE = 4x – 2, and m∠ADE = 75, find: a) the value of x_________ b) DE ________ c) BC_________ d) m∠ABC

D is the midpoint of AB and E is the midpoint of BC: 1. If DE = 8, find AC.

2. If DE = 6, find AC.

5

3. If AC = 20, find DE.

4. If AC = 17, find DE.

5. If BE = 4, find EC.

6. If AD = 7, find DB.

7. If DB = 5, find AB.

8. If BC = 9, find EC.

9. If m∠DBE = 35, find m∠BAC.

9. If m∠BCA = 35, find m∠DEB.

M, R, and T are midpoints of AB, BC and CA, respectively in ΔABC. 13. For each side of ΔABC, name a segment parallel to each side.

14. If AB = 22, BC = 12 and AC = 20, find: a) perimeter of ΔABC______ b) perimeter of ΔMRT______

6

D, E and F are midpoints of RT, TS, and SR respectively in ΔRTS. 15. If DE = 3y – 2 and TS = 4y + 4, find : a) y_____b)DF______c)TS______

16. If FE = x + 3 and RT = 4x – 7, find: a) x_____b)FE______c)RT______

17. If EFDT is a rhombus, EF = 2x – 2 and FD = 4x – 9, find: a) x_____b)EF______c)FD______ d) ST______e)RT______

Proportions Involving Line Segments If a line is parallel to one side of a triangle, and intersects the other two sides, the lines divide those sides proportionally. Example: In triangle RST, a line is drawn parallel to ST intersecting RS in K and RT in L. If RK = 5, KS = 10, and RT = 18, find RL Ex. 1. In triangle RST, a line is drawn parallel to ST intersecting RS in K and RT in L. If RK = 5, KS = 10, and RT = 18, find RL

Ex. 2. In ΔABC, CD = 6, DA = 5, CE = 12, and EB = 10. Is DE parallel AB?

1. If DE ⎪⎪ AC and BD : DA = 3 : 1, find the ratio of BE : EC.

2. If BD = 8, DA = 4, BE = 10, and EC = 5, is DE ⎪⎪ AC.

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3. If AD = 6, BD = 9, EC = 4 and BE = 8, is DE ⎪⎪ AC.

3. If DE ⎪⎪ AC, BD = 6, DA = 2 and BE = 9, find EC.

5. If DE ⎪⎪ AC and BD : DA = 4 : 1 and BE = 40, find EC.

6. If DE ⎪⎪ AC, BE = 5, EC = 3 and BA = 16, find BD.

In ΔRST, DE ⎪⎪ RT 7. If SD = 4, DB = 3 and ST = ST, find SE.

8. If SE = 4, ET = 2, and SR = 9, find SD.

9. If SD = 4, SR = 10, and ST = 5, find SE.

10. If SE = 6, ST = 15, and SR = 20, what is the length of SD.

8

11. If ET = 3, DR = 4, and SR = 12, what is the length of SE.

12. If SD = 12, SR = 30, and ST = 15, find SE.

13. In ΔABC, D is a point on AB, E is a point on AC and De is drawn. AD = 6, DB = 4, and AC = 15. If DE ⎪⎪ BC, find EC.

14. In ΔABC with a line drawn parallel to AC intersecting AB at D and CB at E. If AB = 8, BC = 12 and BD = 6, find BE.

In ΔABC, where D is a point on AC and E is a point on BC such that DE ⎪⎪ AB 15. If CA = 8, AB = 12 and CD = 4, find DE.

16. If CE = 4, DE = 6 and CB = 10, find AB.

17. If AB = 12, DE = 8, and AC = 9, find DC.

18. If CD = 3, DE = 5, and AB = 10, find CA.

9

19. If CD = 4, DE = 4, and DA = 1, find AB.

20. If CD = 3, DA = 2, and AB = 10, find DE.

21. If DE = 4, EB = 2, and DE = 6, find AB.

22. If CD = 6, DE = 8, and AB = 6, find CD.

23. If AB = 9, DE = 6 and EB = 2, find CE.

24. If CD = 8, DE = 8, and DA = 2, find AB.

AC is a diagonal of rectangle ABCD and EF joins the midpoints of AB and BC. 25. If AC = 20, find EF.

26. If EF = 13, and EB = 12, find: a) AC__________ B) AB__________

27. Which one of the following four statements is true? a) EF ⊥ BC b) ΔBEF ΔBAC c) BE = BF d) ΔBEF ~ ΔBAC

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Similar Polygons • Corresponding sides altitudes perimeters or medians are in proportion • Corresponding areas is equal to the square of the sides • Corresponding volumes is equal to the cube of the side

1. The ratio of similitude in two similar triangles is 3 : 1. If a side in the larger triangle measures 30cm, find the measure of the corresponding side in the smaller triangle.

2. If the lengths of the sides of two similar triangles are in the ratio of 5 : 1, find the ratio of the lengths of a pair Of corresponding altitudes, in the order given,

3. The lengths of two corresponding sides of two similar triangles 8 and 12. If an altitude of the smaller triangle has a length of 6, find the length of the corresponding altitude of the larger triangle.

4. The ratio of similitude in two similar triangles is 4 : 3. If the median in the larger triangle measures 12, find the measure of the corresponding median. in

5. The ratio of lengths of the corresponding sides of two triangles is 7 : 4. Find the ratio of the perimeters of the triangles.

6. Corresponding altitudes of two similar triangles have lengths of 9 and 6. If the perimeter of the larger triangle is 24,, find the what the perimeter of the smaller triangle.

11

7. The sides of a triangle are 8, 10, and 12. If the length of the shortest side of a similar triangle is 6, find the length of the longest side.

8. The sides of a triangle measure 7, 9, and 11. Find the perimeter of a similar triangle in which the shortest side Has a length of 21.

9. A vertical pole 10 ft high casts a shadow 8ft long, and at the same time a nearby tree casts a shadow of 40ft long. What is the height of the tree?

10. Triangle DEF is similar to triangle D`E`F`. ∠D corresponds to ∠D` and ∠E corresponds to ∠E`. If DE = 2x + 2, DF = 5x – 7, D`E` = 2 and D`F` = 3, find DE and DF

11. AB represents the width of the river of a river. AE and BD intersect at C. AB ⊥ BD and ED ⊥ BD. If BC = 80, CD = 40 and DE = 20, find AB.

12. In ΔABC, AB ⊥ DE and CB ⊥ AB. If CB = 40, DE = 30 and EB = 20, find AE.

12

1. The lengths of the sides of a triangle are 24, 16, and 12. If the shortest side of a similar triangle is 6, what is the length of the longest side of this triangle.

2. The lengths of the sides of a triangle are 36, 30, and 18. If the longest side of a similar triangle is 9, what is the length of the shortest side of this triangle?

3. A certain tree cast a shadow 6m long. At the same time, a nearby boy 2m tall cast a shadow of 4m long. Find the height of the tree.

4. A building casts a shadow 18 feet long. At the same time, a women 5 feet tall cast a shadow 3 feet long. Find the height of the building.

5. The sides of a triangle measure 18, 20, and 24. If the shortest side of a similar triangle measures 12, find the length of its longest side.

6. A student who is 5 feet tall casts a shadow 8 feet long. At the same time, a tree casts a 40 foot shadow. How many feet is the tree?

13

7. After a 5 by 7 photo is enlarged, its shorter side measures 15 inches. Find the length of the longer side.

8. The sides of a triangle measure 6, 8, and 10. Find the measure of the shortest side of a similar triangle whose perimeter is 16.

9. The measures of two corresponding altitudes of two similar triangles are 6m and 14m. If the perimeter of the first triangle is 21, what is the perimeter of the second triangle.

10. The lengths of the sides of a triangle are 8, 20, and 24. The length of the longest side of a similar triangle is 12. Find the perimeter of the smaller triangle..

11. The lengths of the sides of a triangle are 5, 6, and 7. If the perimeter of a similar triangle is 36 feet, find the shortest length of the other triangle. .

12. The measures of corresponding medians in two similar triangles are in the ratio of 2 : 3. What is the ratio of the area of the smaller triangle to the area of the larger triangle?

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1. If ΔABC ~ ΔDEF and the perimeter of ΔABC is 9, find x, y, and z.

2. . If ΔHIJ ~ ΔKJL and the perimeter of ΔKJL is 7.5, find x, y, and z.

3. If ΔABC ~ ΔA`B`C`, find x and y

4. If ΔABC ~ ΔA`B`C`, find x and y

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5. If ΔABC ~ ΔDEF and the perimeter of ΔDEF is 29, find x, y, and z.

6. If ΔEFG ~ ΔABC and the perimeter of ΔABC is 81, find x, y, and z.

7. The lengths of a triangle measure 24, 16, and 12. If the length of the shortest side has a length of 6, find the length of the longest side.

8. The lengths of a triangle measure 36, 30, and 18. If the length of the longest side has a length of 9, find the length of the shortest side.

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Proportions in Right Triangles with an Altitude Method 1 TOP HAT LESS THAN

a is to b as to c a is to b as to c or or a = b a = b b c b c GREATER THAN

a is to b as to c

or a = b

b c

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Leg2

Method 2

Proportions in a Right Triangle

HLLS/SAAS Method If the altitude is drawn to the hypotenuse of a right triangle, then:

The  two  triangles  formed  are  similar  to  the  given  triangle  and  similar  to  each  other.   ΔACD ~ ΔABC; ΔACD ~ ΔCBD; ΔCBD ~ ΔABC

The  length  of  each  leg  is  the  mean  proportional  between  the  length  of  the  whole  hypotenuse  and  the  length  of  the  segment  adjacent  to  the  leg.    HLLS  

Comparing: ΔABC and ΔACD

AB :AC = AC :ADABAC

=ACAD

AB :BC = BC :BDABBC

=BCBD

HL=LS

The  length  of  the  altitude  is  the  mean  proportional  between  the  lengths  of  the  segments  formed  on  the  hypotenuse.    SAAS  

Comparing: ΔACD and ΔCBD

AD :CD = CD :BDADCD

=CDBD

SA=AS

A B

C

D

Hypotenuse

Leg1

Segment1

Altitude

Segment2

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1. If AD = 3 and CD = 6, find DB.

2. If AB = 8, and AC = 4, find AD.

3. If AC = 10 and AD = 5, find AB.

4. If AC = 6 and AB = 9, find AD.

5. If AD = 4 and DB = 9, find CD.

6. If DB = 4 and BC = 10,, find AB.

7. If AD = 3 and DB = 27, find CD.

8. If AD = 2 and AB = 18, find AC.

9. If DB = 8 and AB = 18, find BC

10. If AD = 3 and DB = 9, find AC.

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11. If AD = 2 and DB = 8, find CD.

12, If AD = 2 and DB = 6, find CD.

13. If CD = 10 and AD = 4, find DB.

14. If CD = 5 and AD = 5, find DB.

15. If AD = 2 and DB = 6, find AC.

16. If AD = 3 and DB = 24, find AC

17. If BC = 10 and AB = 25, find DB.

18. If AC = 4 and DB is 4 more than the length of AD, find CD.

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19. If AD = 12 and DB is three less than the CD, find CD.

20. If AB is 4 times as large as AD and AC is 3 more than AD, find AD.

21. If AD = x + 5, DB = x and CD = 6, find x.

22. If AD = 7, and DB = x and BC = 12, find x.

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23. If AD = x, DB = 12 and AC = 8, find x.

24. If CD = 6, AD = 3 and DB = 5x - 3, find x.

1. If ST ⎪⎪ QR, PS = 4 SQ = 2 and TR = 3. Find PT.

2. If ST ⎪⎪ QR, PT = 16 TR = 8 and PS = 8. Find PQ.

3. If ST ⎪⎪ QR, PR = 12 PS = 6 and TR = 4. Find PQ.

4. If ST ⎪⎪ QR, PQ = 12 SQ = 4 and PT = 10. Find PT.

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5. If ST ⎪⎪ QR, PS = 3 SQ = 3 and PR = 24. Find PT.

6. If ST ⎪⎪ QR, PQ = 10 SQ = 4 and PR = 5. Find PT.

7. If ST ⎪⎪ QR, SQ = 4 , PQ = 12 and TR = 3, find PT.

8. If ST ⎪⎪ QR, PS = 6, PT = 12 and PR = 22. Find SQ.

9. If ST ⎪⎪ QR, PT = 10, PR = 15 and SQ = 4. Find PQ.

10. If ST ⎪⎪ QR, PQ = 15, PS = 10 and PT = 12. Find TR.

23

11. Find x.

12. Find x

13. Find x.

14. Find x.

15. Find x.

16. Find x.

24

17. If DE ⎪⎪ AB, CD = 6, CA = 8, and AB = 12, find DE.

18. If DE ⎪⎪ AB, CE = 3, CB = 5 and DE = 9, find AB.

19. If DE ⎪⎪ AB, CE = 4, EB = 1 and AB = 10, find DE.

20. If DE ⎪⎪ AB, CE = 12, EB = 3 and AB = 30, find DE.

Special Right Triangles 30-60-Right Triangle 45-45-Isosceles Right Triangle

25

1.

2. 3.

4.

5. 6.

7.

8. 9.

10.

11. 12.

26

13.

14. 15.

16.

17. 18.

19.

20. 21.

1. In right ΔABC, ∠C is a right angle and m∠A = 30. If BC = 12, find AC and AB.

2. Find the length of a side of an equilateral triangle whose altitude is .

27

3. What is the length of an altitude of an equilateral triangle whose side is 2?

4. What is the length of an altitude of an equilateral triangle whose side is 9?

5. Each base angle of an isosceles triangle has a measure of 30. If the length of the base is 30, what is the height of the triangle?

6. The perimeter of an equilateral triangle is 12. What is the length of each altitude of the triangle/

7. In a rhombus which contains an angle of 60, the length of each side is 6. Find the length of each diagonal.

8. The length of the hypotenuse of an isosceles right triangle is 10. Find, in simplest radical form, the length of a leg of the triangle.

9. Find, in simplest radical form, the length of a diagonal of a square whose side is 7.

10. If the perimeter of a square is 8, what is the length of the diagonal, in simplest radical form.

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11. If the diagonal of a square has a length of , find the perimeter.

12. If the diagonal of a square has a length of10, find the perimeter, , in simplest radical form.

13. The lengths of the bases of an isosceles trapezoid are 9 and 17. Each leg makes an angle of 45 with the longer base. Find the length of the trapezoid.

14. The lengths of the bases of an isosceles trapezoid are 8 and 18. Each leg makes an angle of 45 with the longer base. Find the length of the trapezoid.

15. In the accompanying diagram of trapezoid ABCD, AD ⊥ DC, AB = 6, DC = 9, and CB = 5. Find AD.

16. In the accompanying diagram, ABCD is an isosceles trapezoid, AD = BC + 5, AB = 10, and DC = 18. Find AE.

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17. If CB = 12, what is the value of AC.?

18. If CB = 4, what is the value of AC?

19. In ΔABC, ∠B is a right angle, DC = DA, m∠CDB = 60, and DB = 5, find: a) m∠CDB______b) m∠CDB______c)DC______ d) m∠CDB_____e) m∠CAD_______f)CB______ g) DA_______ h) AC________ i) AB________

20. ΔABC is right triangle with altitude CD drawn to hypotenuse AB, m∠B = 60, and m∠A = 30. If DB = 2, find: a) CB_______ b) AB________ c) AD________ d) CD_______ e) AC________

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21. ΔABC is right triangle with altitude CD drawn to hypotenuse AB, m∠B = 60, and m∠A = 30. If CB = 6, find: a) AB_______ b) DB________ c) AD________ d) CD_______ e) AC________

22. ΔABC is right triangle with altitude CD drawn to hypotenuse AB, m∠B = 60, and m∠A = 30. If AB = 4, find: a) CB_______ b) DB________ c) AD________ d) CD_______ e) AC________

23. ΔABC is right triangle with altitude CD drawn to hypotenuse AB, m∠B = 60, and m∠A = 30. If AC = find: a) CB_______ b) AB________ c) DB________ d) AD_______ e) CD________

24. ΔABC is right triangle with altitude CD drawn to hypotenuse AB, m∠B = 60, and m∠A = 30. If CD = , find: a) DB_______ b) BC________ c) AB________ d) AD_______ e) AC________

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Review: 1. B is a point on AC such that AB : BC = 2: 3

a) Find the ratio BC : AB_______ b) Find the ratio of AB : AC c) If BC = 6, find AB________ d) If AB = 6, find AC________

2. In ΔABC, D is the midpoint of AB, E is the midpoint of BC, and DE is drawn.

a) Is DE ⎪⎪ AB_____b) If DE = 8, find AC_____ c) If m∠BDE = 70, find m∠BAC_______

3. ΔABC is given where D is a point on AC, and E is a point on CB, such that DE ⎪⎪ AB. If CD = 8, DA = 4 and CE = 6, find EB.

4. ΔABC is given where D is a point on AC, and E is a point on CB, such that DE ⎪⎪ AB. If CE = 10, CB = 15, and CA = 12, find CD.

5. ΔABC is given where D is a point on AC, and E is a point on CB, such that DE ⎪⎪ AB. If CD = 6, DA = 3 and CB = 6, find CE.

6. ΔABC is given where D is a point on AC, and E is a point on CB, such that DE ⎪⎪ AB. If CE = 4, ED = 4 and BA = 6, find CB.

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7. ΔABC is given where D is a point on AC, and E is a point on CB, such that DE ⎪⎪ AB. If CE = 6, ED = 6, and EB = 3, find BA.

8. ΔABC is given where D is a point on AC, and E is a point on CB, such that DE ⎪⎪ AB. If CD = 8, DA = 4 and AB = 15, find DE.

9. ΔABC is given where D is a point on AC, and E is a point on CB, such that DE ⎪⎪ AB. If CE = 6, EB = 3 and DE = 8, find AB.

10. ΔABC is given where D is a point on AC, and E is a point on CB, such that DE ⎪⎪ AB. If CD = 12, DA = 6 and CB = 12, find CE.

11. ΔABC is a right triangle and CD is the altitude drawn to hypotenuse AB. If AD = 18, and DB = 2, find CD.

12. ΔABC is a right triangle and CD is the altitude drawn to hypotenuse AB. If DB = 4, and BC = 8, find AB.

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13. ΔABC is a right triangle and CD is the altitude drawn to hypotenuse AB. If AC = 12 and AB = 18, find AD.

14. ΔABC is a right triangle and CD is the altitude drawn to hypotenuse AB. If AD = 20 and CD = 10, find DB.

15. In rhombus ABCD, diagonals AC and BD intersect at E. The perimeter of the rhombus is 80. and m∠ABC = 60. Find: a) AB_______ b) m∠AEB________ c) m∠DAB________ d) m∠DCA________ e) DE_______ f) DB_________ g) m∠ABD________ h) AC___________ 16. In a right triangle ABC, the length of hypotenuse AC is 5. If BC exceeds AB by 1, find the lengths of AB and BC.

17. In a rectangle, the length is 7 more than the width. The diagonal of the rectangle is 8 more than the width. If x represents the width, write and solve an algebraic equation to find the different dimensions of the rectangle.

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Regent Questions: 1. In the diagram below of , ,

, and .

What is the length of ? 1) 2) 3) 4)

2. An overhead view of a revolving door is shown in the accompanying diagram. Each panel is 1.5 meters wide.

What is the approximate width of d, the opening from B to C? 1) 1.50 m 2) 1.73 m 3) 3.00 m 4) 2.12 m

3. Which set of numbers does not represent the sides of a right triangle? 1) 2) 3) 4)

ANS

4. Which set of numbers could not represent the lengths of the sides of a right triangle? 1) 2) 3) 4)

5. The set of integers is a Pythagorean triple. Another such set is 1) 2) 3) 4)

6. Which set of numbers could be the lengths of the sides of a right triangle? 1) 2) 3) 4)

35

7. The diagram below shows a pennant in the shape of an isosceles triangle. The equal sides each measure 13, the altitude is , and the base is 2x.

What is the length of the base? 1) 5 2) 10 3) 12 4) 24

8. As shown in the diagram below, a kite needs a vertical and a horizontal support bar attached at opposite corners. The upper edges of the kite are 7 inches, the side edges are x inches, and the vertical support bar is inches.

What is the measure, in inches, of the vertical support bar? 1) 23 2) 24 3) 25 4) 26

36

10. In the diagram below of , , , , and .

What is the length of ? 1)

2)

3) 14 4) 24

11. In the diagram below of , .

If , , and , what is the length of

? 1) 5 2) 14 3) 20 4) 26

12. In the diagram of shown below, .

If , , and , what is the length of

? 1) 6 2) 2 3) 3 4) 15

13. In the accompanying diagram of equilateral triangle ABC, and .

If AB is three times as long as DE, what is the perimeter of quadrilateral ABED? 1) 20 2) 30 3) 35 4) 40

37

14. In , point D is on , and point E is on such that . If , , and , what is the length of ? 1) 8 2) 9 3) 10.5 4) 13.5

15. In the diagram below of , D is a point on , E is a point on , , inches,

inches, and inches. Find, to the nearest tenth of an inch, the length of .

16. In the diagram below of , E is a point on and B is a point on , such that . If

, , and , find the length of .

17. In the diagram below of , B is a point on and C is a point on such that ,

, , , and . Find the length of .

38

1. If the midpoints of the sides of a triangle are

connected, the area of the triangle formed is what part of the area of the original triangle? 1)

2)

3)

4)

2. In the diagram below of , is a midsegment of , , , and . Find the perimeter of .

3. In the diagram of below, , , and . Find the perimeter of the triangle formed by connecting the midpoints of the sides of .

4. In the diagram below of , D is the midpoint of , O is the midpoint of , and G is the midpoint of .

If , , and , what is the perimeter of parallelogram CDOG? 1) 21 2) 25 3) 32 4) 40

5. In the diagram of shown below, D is the

midpoint of , E is the midpoint of , and F is the midpoint of .

If , , and , what is the perimeter of trapezoid ABEF? 1) 24 2) 36 3) 40 4) 44

6. In the diagram below, the vertices of are the midpoints of the sides of equilateral triangle ABC, and the perimeter of is 36 cm.

What is the length, in centimeters, of ? 1) 6 2) 12 3) 18 4) 4

39

7. In the diagram below of , D is the midpoint of , and E is the midpoint of .

If , which expression represents DE? 1) 2) 3) 4)

8. In , D is the midpoint of and E is the midpoint of . If and , what is the value of x?

1) 6 2) 7 3) 9 4) 12

9. Triangle ABC is shown in the diagram below.

If joins the midpoints of and , which statement is not true? 1)

2) 3)

4)

10. In the diagram below, joins the midpoints of two sides of .

Which statement is not true? 1)

2)

3) area of area of

4) perimeter of perimeter of