Test on Similarity of triangles

Question 1: The ratio of the perimeters of two similar triangles is equal to their ratio of similarity.

A. Yes

B. No

Explanation: The study materials state, 'If two triangles are similar with a ratio of similarity 'k', then the ratio of their perimeters is also 'k'.' This directly confirms that the ratio of perimeters is equal to the ratio of similarity.

Question 2: If the ratio of similarity 'k' between two similar triangles is 0.5, the second triangle is an enlargement of the first.

A. Yes

B. No

Explanation: The study materials state that if the ratio of similarity 'k' is greater than 1 (k > 1), the second triangle is an enlargement of the first. Conversely, if 0 < k < 1, it is a reduction. Since 0.5 falls within the range of 0 < k < 1, a ratio of similarity of 0.5 means the second triangle is a reduction, not an enlargement, of the first.

Question 3: Which condition must be met for two triangles to be similar according to the SSS (Side-Side-Side) similarity theorem?

A. Two angles of one triangle are congruent to two angles of another triangle.

B. An angle of one triangle is congruent to an angle of another triangle, and the lengths of the sides including these angles are proportional.

C. The corresponding side lengths of the two triangles are proportional.

D. All three pairs of corresponding angles are equal.

Explanation: The SSS (Side-Side-Side) similarity theorem states that if the corresponding side lengths of two triangles are proportional, then the two triangles are similar. This means that if AB / DE = BC / EF = CA / FD = k for ΔABC and ΔDEF, then the triangles are similar. The other options describe AA similarity, SAS similarity, or the general definition of similar triangles.

Question 4: Which set of conditions, if met, would confirm that ΔABC is similar to ΔDEF using the SAS (Side-Angle-Side) Similarity Theorem?

A. ∠A = ∠D, and AB/DE = BC/EF

B. ∠B = ∠E, and AB/DE = BC/EF

C. ∠C = ∠F, and AB/DE = CA/FD

D. AB/DE = BC/EF and BC/EF = CA/FD

Explanation: The SAS Similarity Theorem states that if an angle of one triangle is congruent to an angle of another triangle, and the lengths of the sides *including* these angles are proportional, then the two triangles are similar. Option 1 presents ∠A and sides AB and BC. The angle A is not included between sides AB and BC. Option 2 presents ∠B and sides AB and BC, where ∠B is indeed the included angle between AB and BC. Similarly, ∠E is included between DE and EF. If ∠B = ∠E and AB/DE = BC/EF, then the conditions for SAS similarity are met. Option 3 presents ∠C and sides AB and CA. The angle C is not included between AB and CA. Option 4 describes conditions for SSS (Side-Side-Side) Similarity, not SAS.

Question 5: In a virtual simulation of similar triangles, if one side of the original triangle is doubled, only the corresponding side of the scaled triangle will double, while the other corresponding sides will remain unchanged.

A. Yes

B. No

Explanation: The study materials state that if you double one side of the original triangle, all corresponding sides of the scaled triangle will also double, maintaining the constant ratio of similarity 'k'.