PYQ Vault

Mathematics · Textbook solutions

Triangles

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 55 questions

6.2 Similar Figures

7 q

Exercise 6.1

Practice · 7
  1. Fill in the blanks using the correct word given in brackets :
    Ex 6.1 Q1 (i)
    All circles are __________ . (congruent, similar)
  2. Ex 6.1 Q1 (ii)
    All squares are __________ . (similar, congruent)
  3. Ex 6.1 Q1 (iii)
    All __________ triangles are similar. (isosceles, equilateral)
  4. Ex 6.1 Q1 (iv)
    Two polygons of the same number of sides are similar, if (a) their corresponding angles are __________ and (b) their corresponding sides are __________ . (equal, proportional)
  5. Give two different examples of pair of
    Ex 6.1 Q2 (i)
    similar figures.
  6. Ex 6.1 Q2 (ii)
    non-similar figures.
  7. Ex 6.1 Q3
    State whether the following quadrilaterals are similar or not:

6.3 Similarity of Triangles

16 q

Solved Examples

Worked · 3
  1. 6.2 Eg.1
    If a line intersects sides AB and AC of a △\triangle ABC at D and E respectively and is parallel to BC, prove that ADAB=AEAC\frac{\text{AD}}{\text{AB}} = \frac{\text{AE}}{\text{AC}} (see Fig. 6.13).
  2. 6.2 Eg.2
    ABCD is a trapezium with AB ∥\parallel DC. E and F are points on non-parallel sides AD and BC respectively such that EF is parallel to AB (see Fig. 6.14). Show that AEED=BFFC\frac{\text{AE}}{\text{ED}} = \frac{\text{BF}}{\text{FC}}.
  3. 6.2 Eg.3
    In Fig. 6.16, PSSQ=PTTR\frac{\text{PS}}{\text{SQ}} = \frac{\text{PT}}{\text{TR}} and ∠\angle PST = ∠\angle PRQ. Prove that PQR is an isosceles triangle.

Exercise 6.2

Practice · 13
  1. In Fig. 6.17, (i) and (ii), DE ∥\parallel BC. Find EC in (i) and AD in (ii).
    Ex 6.2 Q1 (i)
    Find EC in (i).
  2. Ex 6.2 Q1 (ii)
    Find AD in (ii).
  3. E and F are points on the sides PQ and PR respectively of a △\triangle PQR. For each of the following cases, state whether EF ∥\parallel QR :
    Ex 6.2 Q2 (i)
    PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm
  4. Ex 6.2 Q2 (ii)
    PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm
  5. Ex 6.2 Q2 (iii)
    PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm
  6. Ex 6.2 Q3
    In Fig. 6.18, if LM ∥\parallel CB and LN ∥\parallel CD, prove that AMAB=ANAD\frac{\text{AM}}{\text{AB}} = \frac{\text{AN}}{\text{AD}}.
  7. Ex 6.2 Q4
    In Fig. 6.19, DE ∥\parallel AC and DF ∥\parallel AE. Prove that BFFE=BEEC\frac{\text{BF}}{\text{FE}} = \frac{\text{BE}}{\text{EC}}.
  8. Ex 6.2 Q5
    In Fig. 6.20, DE ∥\parallel OQ and DF ∥\parallel OR. Show that EF ∥\parallel QR.
  9. Ex 6.2 Q6
    In Fig. 6.21, A, B and C are points on OP, OQ and OR respectively such that AB ∥\parallel PQ and AC ∥\parallel PR. Show that BC ∥\parallel QR.
  10. Ex 6.2 Q7
    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. (Recall that you have proved it in Class IX).
  11. Ex 6.2 Q8
    Using Theorem 6.2, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side. (Recall that you have done it in Class IX).
  12. Ex 6.2 Q9
    ABCD is a trapezium in which AB ∥\parallel DC and its diagonals intersect each other at the point O. Show that AOBO=CODO\frac{\text{AO}}{\text{BO}} = \frac{\text{CO}}{\text{DO}}.
  13. Ex 6.2 Q10
    The diagonals of a quadrilateral ABCD intersect each other at the point O such that AOBO=CODO\frac{\text{AO}}{\text{BO}} = \frac{\text{CO}}{\text{DO}}. Show that ABCD is a trapezium.

6.4 Criteria for Similarity of Triangles

32 q

Solved Examples

Worked · 5
  1. 6.3 Eg.4
    In Fig. 6.29, if PQ ∥\parallel RS, prove that △\triangle POQ ~ △\triangle SOR.
  2. 6.3 Eg.5
    Observe Fig. 6.30 and then find ∠\angle P.
  3. 6.3 Eg.6
    In Fig. 6.31, OA . OB = OC . OD. Show that ∠\angle A = ∠\angle C and ∠\angle B = ∠\angle D.
  4. 6.3 Eg.7
    A girl of height 90 cm is walking away from the base of a lamp-post at a speed of 1.2 m/s. If the lamp is 3.6 m above the ground, find the length of her shadow after 4 seconds.
  5. 6.3 Eg.8
    In Fig. 6.33, CM and RN are respectively the medians of △\triangle ABC and △\triangle PQR. If △\triangle ABC ~ △\triangle PQR, prove that : (i) △\triangle AMC ~ △\triangle PNR (ii) CMRN=ABPQ\frac{\text{CM}}{\text{RN}} = \frac{\text{AB}}{\text{PQ}} (iii) △\triangle CMB ~ △\triangle RNQ

Exercise 6.3

Practice · 27
  1. State which pairs of triangles in Fig. 6.34 are similar. Write the similarity criterion used by you for answering the question and also write the pairs of similar triangles in the symbolic form :
    Ex 6.3 Q1 (i)
    Pair (i)
  2. Ex 6.3 Q1 (ii)
    Pair (ii)
  3. Ex 6.3 Q1 (iii)
    Pair (iii)
  4. Ex 6.3 Q1 (iv)
    Pair (iv)
  5. Ex 6.3 Q1 (v)
    Pair (v)
  6. Ex 6.3 Q1 (vi)
    Pair (vi)
  7. Ex 6.3 Q2
    In Fig. 6.35, △\triangle ODC ~ △\triangle OBA, ∠\angle BOC = 125∘125^\circ and ∠\angle CDO = 70∘70^\circ. Find ∠\angle DOC, ∠\angle DCO and ∠\angle OAB.
  8. Ex 6.3 Q3
    Diagonals AC and BD of a trapezium ABCD with AB ∥\parallel DC intersect each other at the point O. Using a similarity criterion for two triangles, show that OAOC=OBOD\frac{\text{OA}}{\text{OC}} = \frac{\text{OB}}{\text{OD}}.
  9. Ex 6.3 Q4
    In Fig. 6.36, QRQS=QTPR\frac{\text{QR}}{\text{QS}} = \frac{\text{QT}}{\text{PR}} and ∠\angle 1 = ∠\angle 2. Show that △\triangle PQS ~ △\triangle TQR.
  10. Ex 6.3 Q5
    S and T are points on sides PR and QR of △\triangle PQR such that ∠\angle P = ∠\angle RTS. Show that △\triangle RPQ ~ △\triangle RTS.
  11. Ex 6.3 Q6
    In Fig. 6.37, if △\triangle ABE ≅\cong △\triangle ACD, show that △\triangle ADE ~ △\triangle ABC.
  12. In Fig. 6.38, altitudes AD and CE of △\triangle ABC intersect each other at the point P. Show that:
    Ex 6.3 Q7 (i)
    △\triangle AEP ~ △\triangle CDP
  13. Ex 6.3 Q7 (ii)
    △\triangle ABD ~ △\triangle CBE
  14. Ex 6.3 Q7 (iii)
    △\triangle AEP ~ △\triangle ADB
  15. Ex 6.3 Q7 (iv)
    △\triangle PDC ~ △\triangle BEC
  16. Ex 6.3 Q8
    E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that △\triangle ABE ~ △\triangle CFB.
  17. In Fig. 6.39, ABC and AMP are two right triangles, right angled at B and M respectively. Prove that:
    Ex 6.3 Q9 (i)
    △\triangle ABC ~ △\triangle AMP
  18. Ex 6.3 Q9 (ii)
    CAPA=BCMP\frac{\text{CA}}{\text{PA}} = \frac{\text{BC}}{\text{MP}}
  19. CD and GH are respectively the bisectors of ∠\angle ACB and ∠\angle EGF such that D and H lie on sides AB and FE of △\triangle ABC and △\triangle EFG respectively. If △\triangle ABC ~ △\triangle FEG, show that:
    Ex 6.3 Q10 (i)
    CDGH=ACFG\frac{\text{CD}}{\text{GH}} = \frac{\text{AC}}{\text{FG}}
  20. Ex 6.3 Q10 (ii)
    △\triangle DCB ~ △\triangle HGE
  21. Ex 6.3 Q10 (iii)
    △\triangle DCA ~ △\triangle HGF
  22. Ex 6.3 Q11
    In Fig. 6.40, E is a point on side CB produced of an isosceles triangle ABC with AB = AC. If AD ⊥\perp BC and EF ⊥\perp AC, prove that △\triangle ABD ~ △\triangle ECF.
  23. Ex 6.3 Q12
    Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of △\triangle PQR (see Fig. 6.41). Show that △\triangle ABC ~ △\triangle PQR.
  24. Ex 6.3 Q13
    D is a point on the side BC of a triangle ABC such that ∠\angle ADC = ∠\angle BAC. Show that CA2=CB⋅CD\text{CA}^2 = \text{CB} \cdot \text{CD}.
  25. Ex 6.3 Q14
    Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR. Show that △\triangle ABC ~ △\triangle PQR.
  26. Ex 6.3 Q15
    A vertical pole of length 6 m casts a shadow 4 m long on the ground and at the same time a tower casts a shadow 28 m long. Find the height of the tower.
  27. Ex 6.3 Q16
    If AD and PM are medians of triangles ABC and PQR, respectively where △\triangle ABC ~ △\triangle PQR, prove that ABPQ=ADPM\frac{\text{AB}}{\text{PQ}} = \frac{\text{AD}}{\text{PM}}.