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Learn/ 9th Mathematics/ Unit 9 /Numericals

Unit 9: Similar Figures — Exercise 9 2

9th Class Mathematics · Unit 9: Similar Figures

9.2.1.Find the ratio of the areas of similar figures if the ratio of their corresponding lengths are: (i) 1:3 (ii) 3:4 (iii) 2:7 (iv) 8:9 (v) 6:5
Formula
A1A2=left(l1l2right)2
(i) left(frac13right)2=frac19 ⇒ A1:A2=1:9
(ii) left(frac34right)2=916 ⇒ A1:A2=9:16
(iii) left(frac27right)2=449 ⇒ A1:A2=4:49
(iv) left(frac89right)2=6481 ⇒ A1:A2=64:81
(v) left(frac65right)2=3625 ⇒ A1:A2=36:25
(i)1:9 (ii)9:16 (iii)4:49 (iv)64:81 (v)36:25
9.2.2.Find the unknowns in the following figures: (i) A rectangle has A1 = 240 cm² with side 10 cm; a similar rectangle has side 6 cm; find A2. (ii) A triangle has A1 = 60 cm² with side 15 cm; a similar triangle has side 20 cm; find A2. (iii) A trapezoid has side 3.6 cm; a similar trapezoid has A2 = 18 cm² with side 5.76 cm; find A1. (iv) A cone has slant length 15 cm; a similar cone has slant length 12 cm and A2 = 96 cm²; find A1. (v) Two similar triangles (bow-tie figure) have A1 = 3⁴⁄₇ cm² with side 3 cm, and A2 = 63 cm²; find the corresponding side l2.
Working
(i) 240A2=left(106right)2=10036 ⇒ A2=240 × 36100=86.4 cm2
(ii) 60A2=left(1520right)2=225400 ⇒ A2=60 × 400225=106.67 cm2
(iii) A118=left(3.65.76right)2=12.9633.1776 ⇒ A1=12.96 × 1833.1776=7.03125 cm2
(iv) A196=left(1512right)2=225144 ⇒ A1=225 × 96144=150 cm2
Given
(v) A1=3tfrac47=3.57, A2=63, l1=3
Working
(v) 3.5763=left(3l2right)2 ⇒ 0.0587=left(3l2right)2 ⇒ 0.239=3l2
Result
(v) l2=30.239=12.55 cm
9.2.3.Given that area of ΔABC = 36 cm² and mAB = 6 cm, mBD = 4 cm. [BC ∥ DE, with B on AD and C on AE.] Find (a) the area of ΔADE (b) the area of trapezium BCED.
Given
l1=moverline{AB}=6 cm, l2=moverline{AD}=moverline{AB}+moverline{BD}=6+4=10 cm
Working
(a) A1A2=left(l1l2right)2 ⇒ 36A2=left(610right)2=36100
Result
(a) A2=100 cm2
(b) Area of trapezium BCED=Area of ΔADE-Area of ΔABC
(b) =100-36=64 cm2
9.2.4.Given that ΔABC and ΔDEF are similar, with a scale factor of k = 3. If the area of ΔABC is 50 cm², find the area of triangle ΔDEF?
Given
k=3, Area of ΔABC=50 cm2
Formula
frac{Area of ΔDEF}{Area of ΔABC}=k2
Working
Area of ΔDEF=(50)(32)=(50)(9)
Result
Area of ΔDEF=450 cm2
9.2.5.Quadrilaterals ABCD and EFGH are similar, with a scale factor of k = 1/4. If the area of quadrilateral ABCD is 64 cm², find the area of quadrilateral EFGH.
Given
k=frac14, Area of ABCD=64 cm2
Working
Area of EFGH=(64)left(frac14right)2=(64)left(116right)
Result
Area of EFGH=4 cm2
9.2.6.The areas of two similar triangles are 16 cm² and 25 cm². What is the ratio of a pair of corresponding sides?
Formula
left(l1l2right)2=A1A2=1625
Result
l1l2=sqrt{1625=frac45 ⇒ l1:l2=4:5
9.2.7.The areas of two similar triangles are 144 cm² and 81 cm². If the base of the large triangle is 30 cm, find the corresponding base of the smaller triangle.
Formula
A1A2=left(l1l2right)2 ⇒ 14481=left(30l2right)2
129=30l2
Result
l2=9 × 3012=22.5 cm
9.2.8.A regular heptagon is inscribed in a larger regular heptagon and each side of the larger heptagon is 1.7 times the side of the smaller heptagon. If the area of the smaller heptagon is known to be 100 cm², find the area of the larger heptagon.
Given
k=1.7, Area smaller heptagon=100 cm2
Working
Area larger heptagon=(100)(1.7)2=(100)(2.89)
Result
Area larger heptagon=289 cm2