Class 10 Maths – Chapter 6: Triangles

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 â†' Two equilateral triangles of different sizes, two circles. (ii) non-similar figures â†' A square and a rhombus, a circle and an ellipse.

Q3: State whether quadrilaterals are similar â†' Not necessarily. For quadrilaterals to be similar, both corresponding angles must be equal AND corresponding sides must be in proportion.

In figure, DE ∥ BC. Find EC if AD=1.5cm, DB=3cm, AE=1cm.

Q2: E and F are points on sides PQ and PR of Î"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 â†' EF is not parallel to QR

Q3: In Î"ABC, LM ∥ CB, AL/LC=AM/MB. Given AL=xâˆ'2, AC=2x, find x. â†' x/2x = (AL)/(LB) â†' solve for x.

Q4: DE ∥ AC and DF ∥ AE. Prove BF/FE = BE/EC. â†' Using BPT twice.

Q7: Prove that a line drawn through mid-point of one side parallel to another side bisects the third side. â†' Using BPT and mid-point theorem.

Q9: ABCD is trapezium with AB ∥ DC. Diagonals intersect at O. Prove AO/BO=CO/DO. â†' Draw OE ∥ AB through O.

State which pairs of triangles are similar. Write the similarity criterion: (i) ∠A=∠P=60°, ∠B=∠Q=80°, ∠C=∠R=40° â†' AAA criterion, Î"ABC∼Î"PQR

(ii) AB/QR=BC/RP=CA/PQ â†' SSS criterion, Î"ABC∼Î"QRP

Q3: Diagonals AC and BD of trapezium ABCD with AB∥DC intersect at O. Prove OA·OD=OB·OC. â†' Î"AOB∼Î"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 Î"ABC∼Î"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. â†' Using similarity of Î"ADC, Î"CDB, and Î"ACB.

If Î"ABC∼Î"DEF, AB=1.2cm, DE=1.4cm. Ratio of areas? â†' 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. â†' Ratio of areas=k²=1 â‡' k=1 â‡' corresponding sides equal.

Q6: Prove that ratio of areas of two similar triangles = ratio of squares of corresponding angle bisectors.

Which sides of triangles are sides of right triangles? (i) 7,24,25 â†' 7²+24²=49+576=625=25² â†' Right angled. (ii) 3,8,6 â†' 9+64=73≠36 â†' Not. (iii) 50,80,100 â†' 2500+6400=8900≠10000 â†' Not.

Q2: PQR is right at P. M is point on QR such that PM⊥QR. Prove PM²=QM·MR. â†' Î"QPM∼Î"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 Î"ABC. OD⊥BC, OE⊥AC, OF⊥AB. Prove AF²+BD²+CE²=AE²+CD²+BF². â†' Apply Pythagoras in each small triangle.

Q10: Guy wire attached to 18m pole, 24m away from base. Wire length? â†' 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? â†' 12²+5²=169 â†' 13 m

Q14: Prove that thrice the sum of squares of sides of triangle = 4 times sum of squares of medians.

Q2: D is point on hypotenuse AC of right Î"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). â†' Mid-point theorem + similarity.