Exercise 6.1 3 Questions – Basic Concepts
Fill in the blanks using the correct word given in brackets:
(i) All circles are similar (congruent/similar).
(ii) All squares are similar.
(iii) All equilateral triangles are similar (isosceles/equilateral).
(iv) Two polygons with same number of sides are similar if (a) corresponding angles are equal and (b) corresponding sides are proportional.
Q2: Give two different examples of pair of (i) similar figures ' Two equilateral triangles of different sizes, two circles. (ii) non-similar figures ' A square and a rhombus, a circle and an ellipse.
Q3: State whether quadrilaterals are similar ' Not necessarily. For quadrilaterals to be similar, both corresponding angles must be equal AND corresponding sides must be in proportion.
Exercise 6.2 10 Questions – BPT (Thales Theorem)
In figure, DE ∥ BC. Find EC if AD=1.5cm, DB=3cm, AE=1cm.
By BPT: AD/DB = AE/EC ' 1.5/3 = 1/EC ' EC = 3/1.5 = 2 cm
Q2: E and F are points on sides PQ and PR of "PQR. PE=3.9, EQ=3, PF=3.6, FR=2.4. Is EF ∥ QR? Check: PE/EQ=3.9/3=1.3, PF/FR=3.6/2.4=1.5. Not equal ' EF is not parallel to QR
Q3: In "ABC, LM ∥ CB, AL/LC=AM/MB. Given AL=x'2, AC=2x, find x. ' x/2x = (AL)/(LB) ' solve for x.
Q4: DE ∥ AC and DF ∥ AE. Prove BF/FE = BE/EC. ' Using BPT twice.
Q7: Prove that a line drawn through mid-point of one side parallel to another side bisects the third side. ' Using BPT and mid-point theorem.
Q9: ABCD is trapezium with AB ∥ DC. Diagonals intersect at O. Prove AO/BO=CO/DO. ' Draw OE ∥ AB through O.
Exercise 6.3 16 Questions – Similarity Criteria (AAA, SSS, SAS)
State which pairs of triangles are similar. Write the similarity criterion: (i) ∠A=∠P=60°, ∠B=∠Q=80°, ∠C=∠R=40° ' AAA criterion, "ABC∼"PQR
(ii) AB/QR=BC/RP=CA/PQ ' SSS criterion, "ABC∼"QRP
Q3: Diagonals AC and BD of trapezium ABCD with AB∥DC intersect at O. Prove OA·OD=OB·OC. ' "AOB∼"COD using alternate angles.
Q6: Prove that ratio of areas of two similar triangles = ratio of squares of corresponding medians.
Q10: CD and GH are angle bisectors of ∠C and ∠G in "ABC∼"FEG. Prove CD/GH=AC/FG.
Q12: Prove that altitude of equilateral triangle of side a is √3a/2.
Q15: In right triangle ABC with ∠C=90°, D is point on AB such that CD ⊥ AB. Prove CD²=BD·AD. ' Using similarity of "ADC, "CDB, and "ACB.
Exercise 6.4 9 Questions – Areas of Similar Triangles
If "ABC∼"DEF, AB=1.2cm, DE=1.4cm. Ratio of areas? ' ar(ABC)/ar(DEF)=(1.2/1.4)²=(6/7)²=36/49
Q4: Areas of two similar triangles are equal. Prove they are congruent. ' Ratio of areas=k²=1 ' k=1 ' corresponding sides equal.
Q6: Prove that ratio of areas of two similar triangles = ratio of squares of corresponding angle bisectors.
Exercise 6.5 17 Questions – Pythagoras Theorem
Which sides of triangles are sides of right triangles? (i) 7,24,25 ' 7²+24²=49+576=625=25² ' Right angled. (ii) 3,8,6 ' 9+64=73≠36 ' Not. (iii) 50,80,100 ' 2500+6400=8900≠10000 ' Not.
Q2: PQR is right at P. M is point on QR such that PM⊥QR. Prove PM²=QM·MR. ' "QPM∼"PRM.
Q4: ABC is isosceles right triangle with ∠C=90°. Prove AB²=2AC².
Q7: Prove that sum of squares of sides of rhombus = sum of squares of its diagonals.
Q8: O is point inside "ABC. OD⊥BC, OE⊥AC, OF⊥AB. Prove AF²+BD²+CE²=AE²+CD²+BF². ' Apply Pythagoras in each small triangle.
Q10: Guy wire attached to 18m pole, 24m away from base. Wire length? ' L²=18²+24²=900, L=30 m
Q12: Two poles of height 6m and 11m stand on ground. Distance between feet=12m. Distance between tops? ' 12²+5²=169 ' 13 m
Q14: Prove that thrice the sum of squares of sides of triangle = 4 times sum of squares of medians.
Exercise 6.6 (Optional) 10 Questions
Q2: D is point on hypotenuse AC of right "ABC such that BD⊥AC, DM⊥BC, DN⊥AB. Prove DM²=DN·MC.
Q5: D, E, F are mid-points of BC, CA, AB. Prove ar(DEF) = ¼ ar(ABC). ' Mid-point theorem + similarity.
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