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Chapter 1.1

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The document discusses properties of similar figures and how to determine if two figures are similar. It provides examples of similar figures and how to use scale factors and proportional sides to determine missing side lengths. Some key points made include: - Two figures are similar if corresponding angles have the same measure and ratios of corresponding sides are equal. - The scale factor is the ratio of corresponding sides and can be used to determine unknown side lengths of similar figures. - Examples show determining if figures are similar and calculating missing side lengths using scale factors and proportional sides.

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933r Properties of Two Similar Figures Look at the figure of two coins below. Are they the same in shape? Are they the same in size? Figure 1.1 11..11 • scale factor • similar Key terms: What are you going to learn? to decide two similar or dissimilar plane figures by mentioning the properties to determine the length of unknown side of two similar figures Look at the following shapes. Are they the same or different? a. b. c. d. There are two figures in each set. Figures (b) and (c) are different in shape. While, both (a) and (d) are the same in shape, but different in size. Two figures which are the same in shape but different in size are said to be similar figures. Mathematics for Junior High School Grade 9 / 1
What are the properties of two similar figures? Are quadrilaterals ABCD and EFGH below similar? 8 cm A 9 cm B 6 cm D 7.5 cm C 6 cm 8 cm E 12 cm F H 10 cm G Figure 1.2 The corresponding angles of quadrilaterals ABCD and EFGH are the same in size. ∠ A = ∠ E , ∠ B = ∠ F , ∠ C = ∠ G , ∠ D = ∠ H. The corresponding sides are proportional. 4 3 ==== HG DC FG BC EF AB EH AD or 3 4 ==== DC HG BC FG AB EF AD EH . Because the corresponding angles have the same measure and the corresponding sides are proportional, quadrilaterals ABCD and EFGH are similar. Two figures are similar if the corresponding angles have the same measure and the ratio of the lengths of corresponding sides is equal. Look at the following two rectangles. One rectangle is 12 cm in length and 8 cm in width. The other one is 6 cm in length and 4 cm in width. Are they the same? 12 cm 8 cm 6 cm 4 cm Figure 1.3 2 / Student’s Book – Similarity and Congruency
All the angles of the previous rectangles are right angles, so the corresponding angles have the same measure. The ratio of the length is 2 6 12 = , and The ratio of the width is 2 4 8 = . Since they satisfy the properties of similarity of two planes, the two figures are similar. Look at the following triangles. Examine each corresponding sides. Are they proportional? Are the two triangles similar? R 13 Q 5 P 12 L 3 M 5 K 4 The triangle ABC on the right is an isosceles right- angled triangle. AC = 20 cm. Which triangles are similar to ∆EBD? Mathematics for Junior High School Grade 9 / 3
The Length of an Unknown Side of Two Similar Figures Observe the two similar polygons ABCDE and RSTUV below. T B A R S U V CD E 9 x 6 4 5y Figure 1.4 a What is the scale factor of polygon ABCDE to polygon RSTUV? b What are the values of x and y? Solution: a The scale factor is the ratio of the lengths of two corresponding sides. This means that: 9 3 6 2 AE RV = = b Use ratio of corresponding sides to find x and y. 6 4 9 VR RS EA AB x = ⇔ = ⇔ 6x = 36 ⇔ x = 6 and 6 5 9 VR UT EA DC y = ⇔ = ⇔ 6y = 45 ⇔ y = 7.5 4 / Student’s Book – Similarity and Congruency
The three rectangles below are similar. Find the values of x and y. E F G H 7.5 x y 10 A B C D 8 3 1. For the following statements, circle T if the statement is always true, S if it is sometimes true and F if it is always false. a. T S F Two rectangles are similar. b. T S F Two squares are similar. c. T S F A triangle and a quadrilateral are similar. d. T S F Two parallelograms are similar. e. T S F Two equilateral triangles are similar. f. T S F Two rhombuses are similar. g. T S F Two regular pentagons are similar. h. T S F Two isosceles triangles are similar. i. T S F Two kites are similar. 2. Quadrilaterals RSTV and LMNO are similar. The sides of RSTV are 6 cm, 10 cm, 12 cm, and 14 cm in length, respectively. The shortest side of LMNO is 9 cm. a. Determine the scale factor of RSTV to LMNO. b. Find the length of other side of LMNO. c. Find the perimeter of LMNO. d. Find the ratio of the perimeter of RSTV to that of LMNO Mathematics for Junior High School Grade 9 / 5
3. Trapeziums ABCD and AEFG are similar. ∠ AGF = 108o, GF = 14, AD = 12, DG = 4.5, EF = 8 and AB = 26. Determine the scale factor of ABCD to AEFG. Find: a. (i) AG (ii) DC (iii) ∠ADC (iv) BC b. the perimeter of ABCD c. the perimeter of AEFG d. the ratio of the perimeter of ABCD to that of EFGA. 4. The two polygons below are similar. Find the values of x and y. 5. Reasoning Can you give an example of two dissimilar quadrilaterals whose corresponding sides are proportional? 6. Reasoning Can you give an example of two dissimilar quadrilaterals whose corresponding angles are the same size? 7. Reasoning Check whether two triangles whose corresponding sides are proportional are similar. 12 A 15 B P Q x RS C T 12 15 E y D 24 16 E FG BA D C 6 / Student’s Book – Similarity and Congruency
8. Open Question The picture and its frame on the right are similar. The length of the picture is 80 cm, the length of the frame is 100 cm, and the width of the picture is 60 cm. Find the width of the frame. 9. If the three pictures below are similar, determine the values of x and y. 3 6 x y 4 6 10. Quadrilaterals ABCD and RSTU below are similar. Find the values of x, y, and z. D z C 2 4 3 x U T 6 5 y BA R S 11. Quadrilaterals RSTU and WXYZ below are similar. Find the values of a, b, c, and d. U 4 10 T S R ˚ ˚ b 85˚ 108 95 6 W a˚ X 6 d ˚ ˚ 85˚ 108 95 c 3 Y Z Mathematics for Junior High School Grade 9 / 7
12. Open Question A rectangular frame of photograph is 40 cm x 60 cm, and a rectangular photograph is 30 cm x 40 cm. Are the frame and the photograph similar? Suppose we modify the size of the frame so that the frame and the photograph are similar. What is the size? 13. A rectangular folder is 25 cm x 35 cm and a piece of paper is 21 x 32 cm. Are the folder and the paper similar? Modify the size of the paper so that the paper and the folder are similar. 8 / Student’s Book – Similarity and Congruency

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Chapter 1.1

  • 1. 933r Properties of Two Similar Figures Look at the figure of two coins below. Are they the same in shape? Are they the same in size? Figure 1.1 11..11 • scale factor • similar Key terms: What are you going to learn? to decide two similar or dissimilar plane figures by mentioning the properties to determine the length of unknown side of two similar figures Look at the following shapes. Are they the same or different? a. b. c. d. There are two figures in each set. Figures (b) and (c) are different in shape. While, both (a) and (d) are the same in shape, but different in size. Two figures which are the same in shape but different in size are said to be similar figures. Mathematics for Junior High School Grade 9 / 1
  • 2. What are the properties of two similar figures? Are quadrilaterals ABCD and EFGH below similar? 8 cm A 9 cm B 6 cm D 7.5 cm C 6 cm 8 cm E 12 cm F H 10 cm G Figure 1.2 The corresponding angles of quadrilaterals ABCD and EFGH are the same in size. ∠ A = ∠ E , ∠ B = ∠ F , ∠ C = ∠ G , ∠ D = ∠ H. The corresponding sides are proportional. 4 3 ==== HG DC FG BC EF AB EH AD or 3 4 ==== DC HG BC FG AB EF AD EH . Because the corresponding angles have the same measure and the corresponding sides are proportional, quadrilaterals ABCD and EFGH are similar. Two figures are similar if the corresponding angles have the same measure and the ratio of the lengths of corresponding sides is equal. Look at the following two rectangles. One rectangle is 12 cm in length and 8 cm in width. The other one is 6 cm in length and 4 cm in width. Are they the same? 12 cm 8 cm 6 cm 4 cm Figure 1.3 2 / Student’s Book – Similarity and Congruency
  • 3. All the angles of the previous rectangles are right angles, so the corresponding angles have the same measure. The ratio of the length is 2 6 12 = , and The ratio of the width is 2 4 8 = . Since they satisfy the properties of similarity of two planes, the two figures are similar. Look at the following triangles. Examine each corresponding sides. Are they proportional? Are the two triangles similar? R 13 Q 5 P 12 L 3 M 5 K 4 The triangle ABC on the right is an isosceles right- angled triangle. AC = 20 cm. Which triangles are similar to ∆EBD? Mathematics for Junior High School Grade 9 / 3
  • 4. The Length of an Unknown Side of Two Similar Figures Observe the two similar polygons ABCDE and RSTUV below. T B A R S U V CD E 9 x 6 4 5y Figure 1.4 a What is the scale factor of polygon ABCDE to polygon RSTUV? b What are the values of x and y? Solution: a The scale factor is the ratio of the lengths of two corresponding sides. This means that: 9 3 6 2 AE RV = = b Use ratio of corresponding sides to find x and y. 6 4 9 VR RS EA AB x = ⇔ = ⇔ 6x = 36 ⇔ x = 6 and 6 5 9 VR UT EA DC y = ⇔ = ⇔ 6y = 45 ⇔ y = 7.5 4 / Student’s Book – Similarity and Congruency
  • 5. The three rectangles below are similar. Find the values of x and y. E F G H 7.5 x y 10 A B C D 8 3 1. For the following statements, circle T if the statement is always true, S if it is sometimes true and F if it is always false. a. T S F Two rectangles are similar. b. T S F Two squares are similar. c. T S F A triangle and a quadrilateral are similar. d. T S F Two parallelograms are similar. e. T S F Two equilateral triangles are similar. f. T S F Two rhombuses are similar. g. T S F Two regular pentagons are similar. h. T S F Two isosceles triangles are similar. i. T S F Two kites are similar. 2. Quadrilaterals RSTV and LMNO are similar. The sides of RSTV are 6 cm, 10 cm, 12 cm, and 14 cm in length, respectively. The shortest side of LMNO is 9 cm. a. Determine the scale factor of RSTV to LMNO. b. Find the length of other side of LMNO. c. Find the perimeter of LMNO. d. Find the ratio of the perimeter of RSTV to that of LMNO Mathematics for Junior High School Grade 9 / 5
  • 6. 3. Trapeziums ABCD and AEFG are similar. ∠ AGF = 108o, GF = 14, AD = 12, DG = 4.5, EF = 8 and AB = 26. Determine the scale factor of ABCD to AEFG. Find: a. (i) AG (ii) DC (iii) ∠ADC (iv) BC b. the perimeter of ABCD c. the perimeter of AEFG d. the ratio of the perimeter of ABCD to that of EFGA. 4. The two polygons below are similar. Find the values of x and y. 5. Reasoning Can you give an example of two dissimilar quadrilaterals whose corresponding sides are proportional? 6. Reasoning Can you give an example of two dissimilar quadrilaterals whose corresponding angles are the same size? 7. Reasoning Check whether two triangles whose corresponding sides are proportional are similar. 12 A 15 B P Q x RS C T 12 15 E y D 24 16 E FG BA D C 6 / Student’s Book – Similarity and Congruency
  • 7. 8. Open Question The picture and its frame on the right are similar. The length of the picture is 80 cm, the length of the frame is 100 cm, and the width of the picture is 60 cm. Find the width of the frame. 9. If the three pictures below are similar, determine the values of x and y. 3 6 x y 4 6 10. Quadrilaterals ABCD and RSTU below are similar. Find the values of x, y, and z. D z C 2 4 3 x U T 6 5 y BA R S 11. Quadrilaterals RSTU and WXYZ below are similar. Find the values of a, b, c, and d. U 4 10 T S R ˚ ˚ b 85˚ 108 95 6 W a˚ X 6 d ˚ ˚ 85˚ 108 95 c 3 Y Z Mathematics for Junior High School Grade 9 / 7
  • 8. 12. Open Question A rectangular frame of photograph is 40 cm x 60 cm, and a rectangular photograph is 30 cm x 40 cm. Are the frame and the photograph similar? Suppose we modify the size of the frame so that the frame and the photograph are similar. What is the size? 13. A rectangular folder is 25 cm x 35 cm and a piece of paper is 21 x 32 cm. Are the folder and the paper similar? Modify the size of the paper so that the paper and the folder are similar. 8 / Student’s Book – Similarity and Congruency