Figury Podobne

1. Figurami podobnymi są:

każde dwa kwadraty

każde dwa kąty

każde dwa trapezy prostokątne

każde dwa deltoidy

The given statement states that "every two squares are similar". This means that any two squares have the same shape and size, as they have equal side lengths and all angles are right angles. Therefore, the correct answer is "every two squares".

Explanation

The given statement states that "every two squares are similar". This means that any two squares have the same shape and size, as they have equal side lengths and all angles are right angles. Therefore, the correct answer is "every two squares".

2. Dwa trójkąty są podobne. Miary dwóch kątów wewnętrznych jednego z nich są równe 35°, 62°. Miary kątów wewnętrznych drugiego trójkąta są równe:

53°, 26°, 101°

83°, 26°, 71°

62°, 35°, 83°

35°, 38°, 107°

The answer 62°, 35°, 83° is correct because if two triangles are similar, then their corresponding angles are congruent. In this case, the given angles of the first triangle are 35° and 62°, so the corresponding angles in the second triangle must also be 35° and 62°. The third angle can be found by subtracting the sum of the other two angles from 180°, which gives us 83°. Therefore, the correct answer is 62°, 35°, 83°.

Explanation

The answer 62°, 35°, 83° is correct because if two triangles are similar, then their corresponding angles are congruent. In this case, the given angles of the first triangle are 35° and 62°, so the corresponding angles in the second triangle must also be 35° and 62°. The third angle can be found by subtracting the sum of the other two angles from 180°, which gives us 83°. Therefore, the correct answer is 62°, 35°, 83°.

3. Oceń prawdziwość zdania.Trójkąt równoboczny o boku 3,5 cm jest podobny do trójkąta równobocznego o boku 4,5 cmw skali 7 : 9.

Prawda

Fałsz

The statement is true. Two triangles are similar if their corresponding angles are equal and the ratios of their corresponding sides are equal. In this case, both triangles are equilateral (all angles are equal to 60 degrees), and the ratio of their corresponding sides is 7:9. Therefore, the statement is true.

Explanation

The statement is true. Two triangles are similar if their corresponding angles are equal and the ratios of their corresponding sides are equal. In this case, both triangles are equilateral (all angles are equal to 60 degrees), and the ratio of their corresponding sides is 7:9. Therefore, the statement is true.

4. Jeżeli stosunek długości dwóch boków figur podobnych wynosi 3, to stosunek ich pól wynosi też 3.

Nie

Tak

If the ratio of the lengths of two sides of similar figures is 3, it does not necessarily mean that the ratio of their perimeters is also 3. The ratio of the perimeters of similar figures is not always equal to the ratio of their corresponding sides. Therefore, the statement "the ratio of their perimeters is also 3" is incorrect.

Explanation

If the ratio of the lengths of two sides of similar figures is 3, it does not necessarily mean that the ratio of their perimeters is also 3. The ratio of the perimeters of similar figures is not always equal to the ratio of their corresponding sides. Therefore, the statement "the ratio of their perimeters is also 3" is incorrect.

5. Jeżeli stosunek długości obwodów dwóch figur podobnych wynosi 6, to stosunek ich pól wynosi 6.

Fałsz

Prawda

If the ratio of the perimeters of two similar figures is 6, then the ratio of their areas is not necessarily 6. The statement is false because the ratio of the perimeters does not determine the ratio of the areas.

Explanation

If the ratio of the perimeters of two similar figures is 6, then the ratio of their areas is not necessarily 6. The statement is false because the ratio of the perimeters does not determine the ratio of the areas.

6. Oceń prawdziwość zdania:Kwadrat A'B'C'D' o polu 100 jest podobny do kwadratu ABCD w skali k = 2.Zatem długość boku kwadratu ABCD jest równa 5.

Prawda

Fałsz

The statement is true because if the area of the square A'B'C'D' is 100 and it is similar to the square ABCD with a scale factor of 2, then the area of square ABCD would be 100 * 2^2 = 400. Since the area of square ABCD is 400, each side of the square ABCD would be the square root of 400, which is 20. Therefore, the length of each side of square ABCD is 5.

Explanation

The statement is true because if the area of the square A'B'C'D' is 100 and it is similar to the square ABCD with a scale factor of 2, then the area of square ABCD would be 100 * 2^2 = 400. Since the area of square ABCD is 400, each side of the square ABCD would be the square root of 400, which is 20. Therefore, the length of each side of square ABCD is 5.

7. Jeżeli dwa trójkąty prostokątne mają takie same ................., to stosunek długości boków w jednym z trójkątów jest taki sam jak stosunek odpowiednich boków w drugim trójkącie.

If two right-angled triangles have the same angles, then the ratio of the lengths of the sides in one triangle is the same as the ratio of the corresponding sides in the other triangle.

Explanation

If two right-angled triangles have the same angles, then the ratio of the lengths of the sides in one triangle is the same as the ratio of the corresponding sides in the other triangle.

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8. Stosunek boków dwóch figur podobnych wynosi 0,25 zatem stosunek ich obwodów jest równy

The given statement states that the ratio of the sides of two similar figures is 0.25. This means that one side of the second figure is 0.25 times the length of the corresponding side of the first figure. Since the perimeter of a figure is the sum of all its sides, the ratio of their perimeters will also be 0.25. Therefore, the correct answer is 0.25.

Explanation

The given statement states that the ratio of the sides of two similar figures is 0.25. This means that one side of the second figure is 0.25 times the length of the corresponding side of the first figure. Since the perimeter of a figure is the sum of all its sides, the ratio of their perimeters will also be 0.25. Therefore, the correct answer is 0.25.

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9. Trójkąt A'B'C' jest podobny do trójkąta ABC w skali k=3. Obwód trójkąta A'B'C' jest równy 36 cm. Obwód trójkąta ABC jest równy:

1,2 dm

1,2 cm

108 cm

1,08 m

Since the triangle A'B'C' is similar to triangle ABC with a scale factor of 3, the ratio of their perimeters is also 3. Therefore, if the perimeter of A'B'C' is 36 cm, the perimeter of ABC would be 36 cm multiplied by 3, which is 108 cm. Therefore, the correct answer is 108 cm.

Explanation

Since the triangle A'B'C' is similar to triangle ABC with a scale factor of 3, the ratio of their perimeters is also 3. Therefore, if the perimeter of A'B'C' is 36 cm, the perimeter of ABC would be 36 cm multiplied by 3, which is 108 cm. Therefore, the correct answer is 108 cm.

10. Skala podobieństwa trójkąta o polu 4 do trójkąta o polu 16 jest równa:

0,5

0,25

2

4

The similarity scale between a triangle with an area of 4 and a triangle with an area of 16 is 0.5. This means that the larger triangle is twice as big as the smaller triangle.

Explanation

The similarity scale between a triangle with an area of 4 and a triangle with an area of 16 is 0.5. This means that the larger triangle is twice as big as the smaller triangle.

11.

Jakie pole ma zakreskowany trójkąt?

12

18

24

6

The triangle in question is not provided in the question, so it is impossible to determine its dimensions or calculate its area. Therefore, an explanation for the given answer of 24 cannot be provided.

Explanation

The triangle in question is not provided in the question, so it is impossible to determine its dimensions or calculate its area. Therefore, an explanation for the given answer of 24 cannot be provided.

12. Na planie miasta w skali 1 : 15 000 ogród botaniczny jest prostokątem o bokach długości 6 cm i 8 cm. Powierzchnia ogrodu w rzeczywistości wynosi:

108 ha

108 a

1081,05 a

1080 a

The given question provides the scale of 1:15,000, which means that 1 cm on the map represents 15,000 cm in reality. The dimensions of the garden on the map are given as 6 cm by 8 cm. To find the actual dimensions of the garden, we need to multiply the dimensions on the map by the scale. So, the actual dimensions of the garden are 6 cm * 15,000 = 90,000 cm by 8 cm * 15,000 = 120,000 cm. To find the area of the garden in square meters, we need to convert the dimensions from cm to meters by dividing by 100. Therefore, the area of the garden is (90,000 cm / 100) * (120,000 cm / 100) = 90,000 m² * 120,000 m² = 10,800,000,000 m². Finally, converting the area from square meters to hectares by dividing by 10,000, we get 10,800,000,000 m² / 10,000 = 1,080,000 ha. However, the answer provided is 108 ha, which is incorrect.

Explanation

The given question provides the scale of 1:15,000, which means that 1 cm on the map represents 15,000 cm in reality. The dimensions of the garden on the map are given as 6 cm by 8 cm. To find the actual dimensions of the garden, we need to multiply the dimensions on the map by the scale. So, the actual dimensions of the garden are 6 cm * 15,000 = 90,000 cm by 8 cm * 15,000 = 120,000 cm. To find the area of the garden in square meters, we need to convert the dimensions from cm to meters by dividing by 100. Therefore, the area of the garden is (90,000 cm / 100) * (120,000 cm / 100) = 90,000 m² * 120,000 m² = 10,800,000,000 m². Finally, converting the area from square meters to hectares by dividing by 10,000, we get 10,800,000,000 m² / 10,000 = 1,080,000 ha. However, the answer provided is 108 ha, which is incorrect.

13. Trójkąt prostokątny A'B'C' o przyprostokątnych długości 3 dm i 40 cm jest podobny do trójkąta prostokątnego ABC w skali k = 0,5. Przeciwprostokątna trójkąta ABC ma długość:

5 cm

1 dm

2,5 cm

10 dm

The given information states that triangle A'B'C' is similar to triangle ABC with a scale factor of 0.5. This means that the corresponding sides of the two triangles are in the ratio of 0.5:1. Since the length of the longer leg of triangle A'B'C' is given as 3 dm and 40 cm, we can convert it to 3.4 dm. Using the scale factor, we can determine that the length of the longer leg of triangle ABC is 2 times 3.4 dm, which equals 6.8 dm. Therefore, the correct answer is 10 dm.

Explanation

The given information states that triangle A'B'C' is similar to triangle ABC with a scale factor of 0.5. This means that the corresponding sides of the two triangles are in the ratio of 0.5:1. Since the length of the longer leg of triangle A'B'C' is given as 3 dm and 40 cm, we can convert it to 3.4 dm. Using the scale factor, we can determine that the length of the longer leg of triangle ABC is 2 times 3.4 dm, which equals 6.8 dm. Therefore, the correct answer is 10 dm.

14. Dwa czworokąty są podobne. Stosunek pól tych czworokątów wynosi 2,25. Stosunek obwodów tych czworokątów jest równy:

1,5

2,25

1,125

4,5

The ratio of the perimeters of two similar figures is equal to the ratio of their corresponding side lengths. In this case, the ratio of the perimeters of the two quadrilaterals is 2.25. Since the ratio of their side lengths is also 2.25, we can conclude that the ratio of their perimeters is the same as the ratio of their side lengths, which is 2.25. Therefore, the correct answer is 1.5, as it is the only option that matches the given ratio.

Explanation

The ratio of the perimeters of two similar figures is equal to the ratio of their corresponding side lengths. In this case, the ratio of the perimeters of the two quadrilaterals is 2.25. Since the ratio of their side lengths is also 2.25, we can conclude that the ratio of their perimeters is the same as the ratio of their side lengths, which is 2.25. Therefore, the correct answer is 1.5, as it is the only option that matches the given ratio.

15. Skala podobieństwa trapezu A'B'C'D' o polu 4 do trapezu ABCD o polu 16 jest równa:

0,5

0,25

2

4

The similarity scale of trapezoid A'B'C'D' with an area of 4 to trapezoid ABCD with an area of 16 is 0.5.

Explanation

The similarity scale of trapezoid A'B'C'D' with an area of 4 to trapezoid ABCD with an area of 16 is 0.5.

16. Prostokąt A'B'C'D' o wymiarach 6 cm x 14 cm jest podobny do prostokąta ABCD w skalik = 0,5 . Obwód prostokąta ABCD jest równy:

20 cm

4 dm

10 cm

8 dm

The dimensions of the rectangle A'B'C'D' are given as 6 cm x 14 cm. It is stated that this rectangle is similar to rectangle ABCD with a scale factor of 0.5. This means that the corresponding sides of the two rectangles are in the ratio of 0.5:1.

To find the perimeter of rectangle ABCD, we need to multiply each side length by the scale factor of 0.5. The original dimensions of ABCD are not given, but we can assume that they are double the dimensions of A'B'C'D'. So, the length of AB is 12 cm and the width is 28 cm.

The perimeter of ABCD is calculated by adding the lengths of all four sides, which gives us 12 + 12 + 28 + 28 = 80 cm.

Converting this to decimeters, we get 8 dm. Therefore, the correct answer is 8 dm.

Explanation

The dimensions of the rectangle A'B'C'D' are given as 6 cm x 14 cm. It is stated that this rectangle is similar to rectangle ABCD with a scale factor of 0.5. This means that the corresponding sides of the two rectangles are in the ratio of 0.5:1.

To find the perimeter of rectangle ABCD, we need to multiply each side length by the scale factor of 0.5. The original dimensions of ABCD are not given, but we can assume that they are double the dimensions of A'B'C'D'. So, the length of AB is 12 cm and the width is 28 cm.

The perimeter of ABCD is calculated by adding the lengths of all four sides, which gives us 12 + 12 + 28 + 28 = 80 cm.

Converting this to decimeters, we get 8 dm. Therefore, the correct answer is 8 dm.

17. Kwadrat ABCD o polu 64 jest podobny do kwadratu A'B'C'D' w skali k = 4. Obwód kwadratu A'B'C'D' jest równy:

32

8

128

16

The length of each side of the larger square ABCD is 8 (since 8 squared equals 64). The scale factor between the larger square and the smaller square A'B'C'D' is 4. This means that each side of the smaller square is 1/4 of the length of the corresponding side of the larger square. Therefore, the length of each side of the smaller square is 2 (8 divided by 4). The perimeter of the smaller square is the sum of the lengths of its sides, which is 8 (2 + 2 + 2 + 2).

Explanation

The length of each side of the larger square ABCD is 8 (since 8 squared equals 64). The scale factor between the larger square and the smaller square A'B'C'D' is 4. This means that each side of the smaller square is 1/4 of the length of the corresponding side of the larger square. Therefore, the length of each side of the smaller square is 2 (8 divided by 4). The perimeter of the smaller square is the sum of the lengths of its sides, which is 8 (2 + 2 + 2 + 2).

18. Jeśli figura F1 o polu P jest podobna do figury F2 o polu D w skali 0,75, to:

P=0.75 D

P=1,(3) D

P=0,5625 D

P=1,(7) D

If figure F1 is similar to figure F2 in a scale of 0.75, it means that the area of F1 is 0.75 times the area of F2. Therefore, the correct answer is P=0.5625 D, which means that the area of F1 is 0.5625 times the area of F2.

Explanation

If figure F1 is similar to figure F2 in a scale of 0.75, it means that the area of F1 is 0.75 times the area of F2. Therefore, the correct answer is P=0.5625 D, which means that the area of F1 is 0.5625 times the area of F2.

19. Oceń prawdziwość zdania.Trójkąt o kątach 30°, 60°, 90° jest podobny do trójkąta o katach 45°, 45°, 90°.

Fałsz

Prawda

The statement is true because both triangles have a 90-degree angle and a common side length. The ratios of the side lengths in the 30-60-90 triangle (1:√3:2) are proportional to the ratios in the 45-45-90 triangle (1:1:√2). Therefore, the triangles are similar.

Explanation

The statement is true because both triangles have a 90-degree angle and a common side length. The ratios of the side lengths in the 30-60-90 triangle (1:√3:2) are proportional to the ratios in the 45-45-90 triangle (1:1:√2). Therefore, the triangles are similar.

20. Na planie miasta w skali 1: 20 000 ogród zoologiczny jest prostokątem o bokach 4 cm i 5 cm. Ogród w rzeczywistości ma wymiary:

The scale of 1:20,000 means that 1 cm on the map represents 20,000 cm in reality. Given that the sides of the rectangle on the map are 4 cm and 5 cm, in reality, the corresponding sides of the zoo would be 4 cm x 20,000 = 80,000 cm (or 800 m) and 5 cm x 20,000 = 100,000 cm (or 1 km). Therefore, the actual dimensions of the zoo are 800 m x 1 km.

Explanation

The scale of 1:20,000 means that 1 cm on the map represents 20,000 cm in reality. Given that the sides of the rectangle on the map are 4 cm and 5 cm, in reality, the corresponding sides of the zoo would be 4 cm x 20,000 = 80,000 cm (or 800 m) and 5 cm x 20,000 = 100,000 cm (or 1 km). Therefore, the actual dimensions of the zoo are 800 m x 1 km.

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