Similarity - Class 10 MCQ

For two triangles to be similar by the SAS criterion, one angle must be equal in both triangles, and the lengths of the sides that form this angle must be proportional. This ensures that the triangles are similar in shape, even if they differ in size.

In similar triangles, corresponding angles must be equal. This equality of angles is a fundamental property that helps in establishing the similarity of triangles, as it ensures that the triangles maintain their shape despite any differences in size.

The ratio of the areas of similar triangles is the square of the ratio of their corresponding sides. If the side ratio is 1:5, the area ratio would be \( (1:5)^2 = 1:25 \). This mathematical relationship helps in solving problems related to area and similarity in geometry.

When using a scale factor of 1:100, the area of the model is proportional to the square of the scale factor. Therefore, the area ratio would be \( (1/100)^2 = 1/10,000 \). This illustrates how dramatically area diminishes when creating a model, emphasizing the importance of scale in representation.

The Basic Proportionality Theorem states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally. This theorem is useful in various geometric proofs and applications, as it establishes a relationship between the segments created by the parallel line.

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Therefore, if the area ratio is 9:16, the ratio of the corresponding sides is the square root of that ratio, which is 3:4. This relationship is critical in understanding dimensions and scaling in geometry.

The Angle-Angle (AA) criterion states that if two angles of one triangle are equal to two angles of another triangle, then the two triangles are similar. This is because the third angles must also be equal due to the triangle sum property (the sum of angles in a triangle is always 180 degrees).

To find the corresponding sides of a similar triangle when given a scale factor of 2, you multiply each side of the original triangle by 2. Thus, 3 cm becomes 6 cm, 4 cm becomes 8 cm, and 5 cm becomes 10 cm. This demonstrates how scaling affects side lengths while maintaining the triangle's shape.

Two figures are defined as similar if their corresponding angles are equal and the lengths of their corresponding sides are proportional. This means that even if the figures differ in size, they maintain the same shape, which is a key concept in geometry.

Congruent figures are those that are identical in both shape and size. This means that all corresponding sides and angles are equal, making them superimposable on one another. This is distinct from similar figures, which may vary in size but maintain the same shape.