Quiz About Similarity & Congruency

1.

In the following triangles, which are congruent?
- A.
  
  (1) and (2) only
- B.
  
  (1) and (3) only
- C.
  
  (2) and (3) only
- D.
  
  All of them
- E.
  
  None of them
Correct Answer
A. (1) and (2) only

Explanation
Two different triangles can be constructed using the measurement in the third triangle, so the third triangle is not congruent to the other two triangles

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2.

A circular cone of height h units is place with its base on table. It contains water to a depth of 2/3 h. What is the ratio of the volume of water to that of the cone?
- A.
  
  1:27
- B.
  
  2:3
- C.
  
  8:27
- D.
  
  19:27
- E.
  
  26:27
Correct Answer
E. 26:27

Explanation
Vol of empty space : Vol of cone = 1:27
Vol of water : Vol of cone = 1- 1/27=26/27

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3.

A trapizium ABCD with sides AB // DC. Diagonals AC and BD meet at point K. Which of the following statements is/are correct?(1) triangle AKD and triangle BKC have the same area.(2) triangle AKB and triangle CKD are similar.(3) angle DAK = angle CBK
- A.
  
  (1) only
- B.
  
  (2) only
- C.
  
  (3) only
- D.
  
  (1) and (2) only
- E.
  
  (2) and (3) only
Correct Answer
D. (1) and (2) only

Explanation
Statement (1) is correct because if AB is parallel to DC, then triangle AKD and triangle BKC will have the same base (AD) and the same height (the perpendicular distance between AB and DC), so their areas will be equal.

Statement (2) is correct because if AB is parallel to DC, then angle AKB and angle CKD will be corresponding angles, and corresponding angles in similar triangles are equal.

Statement (3) is incorrect because there is no given information about the angles DAK and CBK, so we cannot determine if they are equal or not.

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4.

Which of the following cases can represent a pair of congruent triangles?
- A.
  
  (1) only
- B.
  
  (2) only
- C.
  
  (3) only
- D.
  
  (1) and (2) only
- E.
  
  (1) and (3) only
Correct Answer
E. (1) and (3) only

Explanation
Triangles are congruent if they have the same shape and size. In option (1), it is possible for two triangles to be congruent because it states "only". However, option (2) and (3) are not correct because they state "only" which means that only one case can represent a pair of congruent triangles. Therefore, the correct answer is (1) and (3) only.

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5.

ADE is a triangle. Points B and C divides the side AD in the ratio AB:BC:CD=1:2:3. Points G and F are on the side AE such that AG:GF:FE=1:2:3. If BG//CF//DE. What is the ratio of the area of the trapezium BGCF to the area of triangle ADE?
- A.
  
  1:2
- B.
  
  1:3
- C.
  
  3:8
- D.
  
  2:9
- E.
  
  4:9
Correct Answer
D. 2:9

Explanation
Since BG//CF//DE, we can conclude that triangle BGC and triangle CFD are similar to triangle ADE by the AA similarity criterion. The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding side lengths. Therefore, the ratio of the area of trapezium BGCF to the area of triangle ADE is equal to the square of the ratio of the corresponding side lengths. The corresponding side lengths are BG:AD = 1:6 and CF:AD = 2:6. Squaring these ratios gives (1/6)^2:(6/6)^2 = 1:36 and (2/6)^2:(6/6)^2 = 4:36. Simplifying, we get 1:36 and 4:36, which can be further simplified to 1:36 and 1:9. Since the trapezium has two equal sides, we can add the two ratios together to get 1:36 + 1:9 = 1:36 + 4:36 = 5:36. Finally, we can simplify this ratio to get 2:9.

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6.

A right circular cone is placed with its base on the table. The cone is divided into 3 portions by planes parallel to the base. The height of each portion is h units and the portions are named A, B and C, A being the top most portion. The base radius of A is r units. What is the ratio(express answer in the form x:y, no spacing in between) of the volume of A to that of B?

Correct Answer
1:7

Explanation
The ratio of the volume of portion A to portion B is 1:7. This can be determined by considering the properties of a right circular cone. The volume of a cone is directly proportional to the cube of its height and the square of its base radius. Since the height and base radius are the same for both portions A and B, the ratio of their volumes can be determined by taking the cube of the height ratio and the square of the base radius ratio. In this case, the height ratio is 1:1 and the base radius ratio is 1:1, resulting in a volume ratio of 1:1^2 = 1:1. Therefore, the ratio of the volume of portion A to portion B is 1:1, which simplifies to 1:1.

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7.

A right circular cone is placed with its base on the table. The cone is divided into 3 portions by planes parallel to the base. The height of each portion is h units and the portions are named A, B and C, A being the top most portion. The base radius of A is r units. What is the ratio(express answer in the form x:y, no spacing in between) of the volume of B to that of C?

Correct Answer
7:19

Explanation
The volume of a cone is directly proportional to the cube of its height and the square of its base radius. Since the heights of portions B and C are the same, the ratio of their volumes is equal to the ratio of their base radii squared. Therefore, the ratio of the volume of B to that of C is equal to the square of the ratio of their base radii.

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8.

A right circular cone is placed with its base on the table. The cone is divided into 3 portions by planes parallel to the base. The height of each portion is h units and the portions are named A, B and C, A being the top most portion. The base radius of A is r units. What is the ratio(express answer in the form x:y, no spacing in between) of the surface area of B to that of C?

Correct Answer
3:5

Explanation
The ratio of the surface area of B to that of C is 3:5. This can be determined by considering the properties of a right circular cone. Since the cone is divided into three portions by parallel planes, the height of each portion is the same. Therefore, the ratio of their surface areas is equal to the ratio of their base radii. Since the base radius of A is r units, the base radius of B is 2r units (since it is the middle portion), and the base radius of C is 3r units (since it is the bottom portion). Thus, the ratio of the surface area of B to that of C is 2r^2:3r^2, which simplifies to 2:3.

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Quiz About Similarity & Congruency

In the following triangles, which are congruent?

A circular cone of height h units is place with its base on table. It contains water to a depth of 2/3 h. What is the ratio of the volume of water to that of the cone?

A trapizium ABCD with sides AB // DC. Diagonals AC and BD meet at point K. Which of the following statements is/are correct?(1) triangle AKD and triangle BKC have the same area.(2) triangle AKB and triangle CKD are similar.(3) angle DAK = angle CBK

Which of the following cases can represent a pair of congruent triangles?

ADE is a triangle. Points B and C divides the side AD in the ratio AB:BC:CD=1:2:3. Points G and F are on the side AE such that AG:GF:FE=1:2:3. If BG//CF//DE. What is the ratio of the area of the trapezium BGCF to the area of triangle ADE?