Question Bank — Triangles

Section A — MCQs (1 mark each)

1. All circles are
   (a) congruent
   (b) similar
   (c) both congruent and similar
   (d) neither congruent nor similar

2. Two polygons of the same number of sides are similar if
   (a) their corresponding angles are equal
   (b) their corresponding sides are proportional
   (c) both (a) and (b)
   (d) either (a) or (b)

3. In \(\triangle ABC\), if DE \(\parallel\) BC with D on AB and E on AC, then
   (a) \(\frac{AD}{DB} = \frac{AE}{EC}\)
   (b) \(\frac{AD}{AB} = \frac{AE}{AC}\)
   (c) both (a) and (b)
   (d) none of these

4. If in two triangles, corresponding angles are equal, then the triangles are similar by
   (a) SSS similarity criterion
   (b) SAS similarity criterion
   (c) AAA similarity criterion
   (d) RHS similarity criterion

5. Assertion (A): All equilateral triangles are similar.
   Reason (R): All equilateral triangles have equal corresponding angles and proportional sides.
   (a) Both A and R are true and R is the correct explanation of A.
   (b) Both A and R are true but R is not the correct explanation of A.
   (c) A is true but R is false.
   (d) A is false but R is true.

6. Assertion (A): If a line divides two sides of a triangle in the same ratio, then it is parallel to the third side.
   Reason (R): This is the converse of the Basic Proportionality Theorem.
   (a) Both A and R are true and R is the correct explanation of A.
   (b) Both A and R are true but R is not the correct explanation of A.
   (c) A is true but R is false.
   (d) A is false but R is true.

7. In \(\triangle ABC\) and \(\triangle DEF\), if \(\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}\), then the triangles are similar by
   (a) AAA criterion
   (b) SSS similarity criterion
   (c) SAS similarity criterion
   (d) AA criterion

8. If \(\triangle ABC \sim \triangle PQR\), then which of the following is correct?
   (a) \(\angle A = \angle P\)
   (b) \(\frac{AB}{PQ} = \frac{BC}{QR}\)
   (c) both (a) and (b)
   (d) none of these

9. A line drawn through the mid-point of one side of a triangle parallel to another side
   (a) bisects the third side
   (b) is parallel to the third side
   (c) forms a congruent triangle
   (d) none of these

10. If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, the triangles are similar by
    (a) AAA criterion
    (b) SSS similarity criterion
    (c) SAS similarity criterion
    (d) AA criterion

Section B — Very Short Answer (2 marks each)

1. State the Basic Proportionality Theorem.

2. If \(\triangle ABC \sim \triangle DEF\), write the correspondence of vertices and the similarity statement.

3. In Fig. 6.17(i) (DE \(\parallel\) BC), if AD = 4 cm, DB = 6 cm and AE = 5 cm, find EC.

4. Give one example each of a pair of similar figures and a pair of non-similar figures.

5. If in \(\triangle ABC\), DE \(\parallel\) BC, D on AB and E on AC, prove that \(\frac{AD}{AB} = \frac{AE}{AC}\).

6. State the condition under which two polygons are similar.

Section C — Short Answer (3 marks each)

1. In \(\triangle PQR\), E and F are points on PQ and PR respectively such that EF \(\parallel\) QR. If PE = 4 cm, EQ = 4.5 cm, PF = 8 cm and RF = 9 cm, state whether EF \(\parallel\) QR. Give reason.

2. Using Theorem 6.1, prove that a line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side.

3. In Fig. 6.36, if \(\frac{QR}{QT} = \frac{QS}{PR}\) and \(\angle 1 = \angle 2\), show that \(\triangle PQS \sim \triangle TQR\).

4. Diagonals AC and BD of trapezium ABCD with AB \(\parallel\) DC intersect at O. Show that \(\frac{OA}{OC} = \frac{OB}{OD}\).

5. In \(\triangle ABC\), if \(\angle A = 80^\circ\), \(\angle B = 60^\circ\) and in \(\triangle PQR\), \(\frac{AB}{RQ} = \frac{BC}{QP} = \frac{CA}{PR}\), find \(\angle P\).

Section D — Long Answer (5 marks each)

1. State and prove the Basic Proportionality Theorem (Thales Theorem).

2. ABCD is a trapezium with AB \(\parallel\) DC. E and F are points on AD and BC respectively such that EF \(\parallel\) AB. Show that \(\frac{AE}{ED} = \frac{BF}{FC}\). (Multi-step problem)

3. Prove that if in two triangles, corresponding angles are equal, then their corresponding sides are proportional (AAA similarity criterion). Hence state the AA similarity criterion.

Section E — Case/Source-Based (4 marks each)

Case 1

A girl of height 90 cm walks away from a lamp-post 3.6 m high at 1.2 m/s. After 4 seconds, find the length of her shadow.

(i) Draw a labelled diagram showing the lamp-post, girl and her shadow.
(ii) Identify two similar triangles and state the similarity criterion used.
(iii) Set up the proportion using corresponding sides.
(iv) Calculate the length of the shadow.

Case 2

A vertical pole 6 m high casts a shadow 4 m long. At the same time a tower casts a shadow 28 m long.

(i) Draw a diagram showing the pole, tower and their shadows.
(ii) Identify the similar triangles and give the reason for similarity.
(iii) Write the proportion of corresponding sides.
(iv) Find the height of the tower.