Moderate Level Worksheet

Very Short Answer Questions (1 Mark Each)

(1) Name any two similarity criteria for triangles.

Two criteria: AA and SSSSAS and RHSSSS and RHS

Perfect! AA (Angle-Angle), SSS (Side-Side-Side), and SAS are the main similarity criteria.

(2) If two triangles are similar and one has a side of 5 cm corresponding to 10 cm in the other, find the ratio of similarity.

Ratio = smaller : larger = 1:22:1

Excellent! The ratio of similarity is 1:2 or 5:10.

(3) If two triangles have sides in the ratio 4:5, what is the ratio of their perimeters?

Since perimeter is sum of all sides, ratio of perimeters =

Correct! Perimeters have the same ratio as corresponding sides.

(4) What is the condition for two right-angled triangles to be similar?

Since both have 90° angles, they need: One more angle equalTwo more angles equalAll sides equalHypotenuse equal

Perfect! Just one more equal angle makes them similar (AA criterion).

Short Answer Questions (2 Marks Each)

(1) In △ABC, DE ∥ BC and intersects AB and AC at D and E. If AD = 2.5 cm, DB = 5 cm, and AE = 3 cm, find EC. EC = cm

Excellent! EC = 6 cm using the Basic Proportionality Theorem.

(2) In △XYZ, point P lies on XY and Q on XZ such that PQ ∥ YZ. Prove that XPPY = XQQZ.

We can prove this by Basic ProportionalityMultiplicative Proportions Theorem

Perfect! This directly follows from the Basic Proportionality Theorem.

(3) In two similar triangles, the corresponding altitudes are 6 cm and 9 cm. If the area of the smaller triangle is 54 cm2, find the area of the larger triangle.

Ratio of altitudes =

Ratio of areas =

Area of larger triangle = cm2

Excellent! Area of larger triangle = 121.5 cm2.

Long Answer Questions (4 Marks Each)

(1) Prove: If in two triangles, the corresponding sides are in the same ratio, then the triangles are similar (SSS Criterion).

(2) In triangle ABC, DE is drawn parallel to BC, intersecting AB and AC at D and E respectively. Prove that triangle ADE is similar to triangle ABC and hence find the ratio of their areas.

Ratio of Areas: Ratio of areas = k2 = ADAB2AD·AEAB·AC2

Perfect! The areas are in the ratio of square of corresponding sides.

(3) Two triangles are similar and the perimeter of the smaller triangle is 36 cm. The perimeter of the larger triangle is 48 cm. If the area of the larger triangle is 96 cm2, find the area of the smaller triangle. cm2

Excellent! Area of smaller triangle = 54 cm2.

Part B: Objective Questions (1 Mark Each)

Choose the correct answer and write the option (a/b/c/d)

(1) If two triangles are similar, then the ratio of their areas is equal to:

(a) The square of the ratio of corresponding sides (b) The ratio of their perimeters (c) The ratio of corresponding medians (d) None of these

Correct! The ratio of areas of similar triangles equals the square of the ratio of corresponding sides.

(2) In triangle PQR, ST is drawn parallel to QR such that it intersects sides PQ and PR at S and T respectively. Then PSSQ = PTTR is due to:

(a) SAS similarity (b) Basic Proportionality Theorem (c) AA similarity (d) None of these

Correct! This is the direct application of the Basic Proportionality Theorem (Thales' theorem).

(3) If two triangles have corresponding sides in the ratio 3:4, then the ratio of their areas is:

(a) 3:4 (b) 9:16 (c) 16:9 (d) 2:3

Correct! Ratio of areas = 3:42 = 9:16.

(4) If two triangles have the same height and their bases are in the ratio 5:6, then their areas will be in the ratio:

(a) 25:36 (b) 6:5 (c) 5:6 (d) 1:1

Correct! Since Area = 12 × base × height, and heights are equal, areas are in the ratio of bases = 5:6.

(5) Which of the following is a correct statement?

(a) All congruent triangles are similar (b) All similar triangles are congruent (c) All right triangles are similar (d) Equilateral triangles are not similar

Correct! Congruent triangles have the same shape and size, so they're always similar (ratio 1:1).

(6) Triangle DEF is similar to triangle XYZ. If ∠D = 30°, ∠E = 90°, then ∠X =

(a) 60° (b) 30° (c) 90° (d) 120°

Correct! In △DEF: ∠F = 180° - 30° - 90° = 60°. Since triangles are similar, corresponding angles are equal, so ∠X = 60°.

(7) The sides of a triangle are 6 cm, 8 cm, and 10 cm. A similar triangle has its smallest side as 3 cm. The longest side of the smaller triangle is:

(a) 6 cm (b) 5 cm (c) 4 cm (d) 3 cm

Correct! Ratio = 3:6 = 1:2. So longest side = 10 × 12 = 5 cm.

(8) If in two triangles, ∠A = ∠D, ∠B = ∠E, then the triangles are similar by:

(a) AA Criterion (b) SAS Criterion (c) SSS Criterion (d) RHS Criterion

Correct! Two angles equal means similarity by AA (Angle-Angle) criterion.

(9) A triangle is divided into two similar triangles by a line parallel to one of its sides. The ratio of the areas of the two triangles is 1:4. The ratio of the corresponding sides is:

(a) 1:2 (b) 2:1 (c) 1:4 (d) 2:3

Correct! If area ratio is 1:4, then side ratio is 1:4 = 1:2.

(10) In triangle ABC, D and E are midpoints of sides AB and AC respectively. Then DE ∥ BC and DE = 12 BC by:

(a) Converse of BPT (b) Midpoint Theorem (c) Thales' Theorem (d) SSS Similarity

Correct! The Midpoint Theorem states that the line joining midpoints of two sides is parallel to the third side and half its length.

Similar Triangles Challenge

Determine whether these statements about similar triangles are True or False:

Similar Triangles Quiz