Hard Level Worksheet
Very Short Answer Questions (1 Mark Each)
(1) State the AA criterion for similarity of triangles. If two
Perfect! AA (Angle-Angle) similarity is one of the fundamental criteria.
(2) What is the relationship between the areas of two similar triangles and the square of the corresponding sides? The ratio of areas of similar triangles equals the
Excellent! If sides are in ratio k:1, then areas are in ratio
(3) If two triangles are similar and the ratio of their areas is 49:64, find the ratio of their corresponding sides.
Correct! Take square root of area ratio to get side ratio.
(4) Fill in the blank: If two triangles are similar, the ratio of their medians is equal to the ratio of their ?
Perfect! All linear measurements scale proportionally in similar triangles.
Short Answer Questions (2 Marks Each)
Note: Answer each question with steps and explanation, in 2-3 sentences. Write down the answers on sheet and submit to the school subject teacher.
(1) In triangles ABC and DEF, it is given that
Yes, the triangles are similar by
Excellent! SSS similarity requires all corresponding sides to be proportional.
(2) Prove that if a perpendicular is drawn from the vertex of the right angle to the hypotenuse in a right triangle, then the triangles on either side of the perpendicular are similar to the original triangle.
(3) In △ABC, DE ∥ BC, intersecting AB and AC at D and E. Prove that
Long Answer Questions (4 Marks Each)
Note: Answer each question with steps and explanation. Write down the answers on sheet and submit to the school subject teacher.
(1) Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
(2) D and E are points on the sides AB and AC of triangle ABC such that DE || BC. Show that the area of triangle ADE is to the area of triangle ABC as
(3) In triangle PQR, a line is drawn parallel to QR intersecting PQ at M and PR at N. Prove that triangle PMN is similar to triangle PQR and find the ratio of their areas if PM = 3 cm and PQ = 5 cm. Areas Ratio =
Perfect numerical calculation!
Part B: Objective Questions (1 Mark Each)
Choose the correct answer and write the option (a/b/c/d)
(1) Two similar triangles have areas in the ratio 1:9. The ratio of the corresponding altitudes is:
(a) 1:9 (b) 1:3 (c) 3:1 (d) 1:6
Correct! If area ratio is 1:9, then side ratio is
(2) In triangle XYZ, a line is drawn parallel to side YZ to intersect sides XY and XZ at points P and Q. Then triangle XPQ is similar to triangle XYZ by:
(a) AA similarity (b) SSS similarity (c) SAS similarity (d) RHS similarity
Correct! Parallel lines create corresponding angles that are equal, giving us AA similarity.
(3) The ratio of the perimeters of two similar triangles is 2:5. What is the ratio of their areas?
(a) 2:5 (b) 4:25 (c) 5:2 (d) 2:25
Correct! Area ratio =
(4) In two similar triangles, the ratio of the areas is 16:25. If the longer side of the smaller triangle is 8 cm, the corresponding side in the larger triangle is:
(a) 10 cm (b) 12.5 cm (c) 15 cm (d) 20 cm
Correct! Side ratio =
(5) Which of the following is not a correct condition for triangle similarity?
(a) Two angles of one triangle are respectively equal to two angles of another triangle
(b) The ratio of three pairs of corresponding sides is equal
(c) The ratio of corresponding altitudes is not equal
(d) One angle is equal and the sides including the angle are in the same ratio
Correct! If triangles are similar, corresponding altitudes must be in the same ratio as sides.
(6) In a triangle, if a line divides two sides in the same ratio and is not parallel to the third side, then:
(a) It is a median (b) It is perpendicular (c) It contradicts BPT (d) None of the above
Correct! Basic Proportionality Theorem states that if a line divides two sides proportionally, it must be parallel to the third side.
(7) A triangle has side lengths 6 cm, 8 cm, and 10 cm. A triangle similar to it has a perimeter of 48 cm. The length of the largest side of the second triangle is:
(a) 20 cm (b) 18 cm (c) 24 cm (d) 16 cm
Correct! Original perimeter = 24 cm. Scale factor =
(8) Triangle ABC and DEF are similar. If AB = 3 cm, DE = 4 cm, and the area of triangle ABC is 27
(a) 48
Correct! Area ratio =
(9) In triangle XYZ, a line drawn from X divides angle YXZ into two equal parts and intersects side YZ at M. If triangle XYM is similar to triangle XZM, then the angle bisector divides the opposite side in the ratio:
(a) 1:1 (b) 2:1 (c) Equal to ratio of adjacent sides (d) Can't say
Correct! By the angle bisector theorem, the angle bisector divides the opposite side in the ratio of the adjacent sides.
(10) Which of the following is true about the medians of two similar triangles?
(a) Their ratio is equal to the ratio of areas
(b) Their ratio is equal to the ratio of corresponding altitudes
(c) Their ratio is equal to the ratio of corresponding sides
(d) None of these
Correct! In similar triangles, all linear measurements (sides, altitudes, medians) are in the same ratio.
Similar Triangles Challenge
Determine whether these statements about similar triangles are True or False:
Similar Triangles Quiz
🎉 You Did It! What You've Learned:
By completing this worksheet, you now have a solid understanding of:
(1) Similarity Criteria: AA, SSS, and SAS conditions for triangle similarity
(2) Area Relationships: Understanding that area ratio equals square of side ratio
(3) Proportionality Theorems: Basic Proportionality Theorem and its applications
(4) Linear Measurements: How sides, altitudes, medians scale proportionally
(5) Geometric Proofs: Proving similarity using angle and side relationships
(6) Parallel Line Properties: How parallel lines create similar triangles
(7) Practical Applications: Calculating unknown measurements using similarity
(8) Problem-solving Strategies: Multiple approaches to similarity problems
Excellent work mastering the advanced concepts of similar triangles and their applications!