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Learn/ 9th Mathematics/ Unit 9 /Numericals

Unit 9: Similar Figures — Review Exercise

9th Class Mathematics · Unit 9: Similar Figures

R9.2.If the sum of the interior angles of a polygon is 1080°, how many sides does the polygon have?
Formula
Sum of interior angles=(n-2) × 180^circ
Working
1080^circ=(n-2) × 180^circ ⇒ 1080^circ180^circ=n-2 ⇒ 6=n-2
Result
n=8
R9.3.Two similar bottles are such that one is twice as high as the other. What is the ratio of their surface areas and their capacities?
Working
(a) A1A2=left(h1h2right)2=left(2x1xright)2=left(frac21right)2=frac41
Result
(a) A1:A2=4:1
Working
(b) V1V2=left(h1h2right)3=left(frac21right)3=frac81
Result
(b) V1:V2=8:1
R9.4.Each dimension of a model car is 1/10 of the corresponding car dimension. Find the ratio of: (a) the areas of their windscreens (b) the capacities of their boots (c) the widths of the cars (d) the number of wheels they have.
Working
(a) A1A2=left(110right)2=1100 ⇒ A1:A2=1:100
(b) V1V2=left(110right)3=11000 ⇒ V1:V2=1:1000
Result
(c) l1l2=110 ⇒ l1:l2=1:10
(d) ratio of wheels of car=frac44=frac11 ⇒ 1:1
R9.5.Three similar jugs have heights 8 cm, 12 cm and 16 cm. If the smallest jug holds 1/2 litre, find the capacities of the other two.
Working
(a) V1V2=left(h1h2right)3 ⇒ 1/2V2=left(812right)3=left(frac23right)3=827
Result
(a) V2=1 × 272 × 8=1.69 liter
Working
(b) V1V3=left(h1h3right)3 ⇒ 1/2V3=left(816right)3=left(frac12right)3=frac18
Result
(b) V3=1 × 82 × 1=4 liter
R9.6.Three similar drinking glasses have heights 7.5 cm, 9 cm and 10.5 cm. If the tallest glass holds 343 millilitres, find the capacities of the other two.
Working
(a) V1V3=left(h1h3right)3=left(7.510.5right)3=left(1.52.1right)3 ⇒ V10.343=3.3759.261
Result
(a) V1=3.375 × 0.3439.261=0.125 liter=125 mL
Working
(b) V2V3=left(h2h3right)3=left(910.5right)3 ⇒ V20.343=7291157.625
Result
(b) V2=729 × 0.3431157.625=0.216 liter=216 mL
R9.7.A toy manufacturer produces model cars which are similar in every way to the actual cars. If the ratio of the door area of the model to the door area of the car is 1 cm to 2500 cm, find: (a) the ratio of their lengths (b) the ratio of the capacities of their petrol tanks (c) the width of the model, if the actual car is 150 cm wide (d) the area of the rear window of the actual car if the area of the rear window of the model is 3 cm².
Working
(a) A1A2=left(l1l2right)2 ⇒ 12500=left(l1l2right)2 ⇒ sqrt{12500=150
Result
(a) l1:l2=1:50
Working
(b) V1V2=left(l1l2right)3=left(150right)3=1125000
Result
(b) V1:V2=1:125000
Working
(c) w1w2=l1l2 ⇒ w1150=150
Result
(c) w1=15050=3 cm
Working
(d) A1A2=left(l1l2right)2 ⇒ 3A2=left(150right)2=12500
Result
(d) A2=3 × 2500=7500 cm2
R9.8.The ratio of the areas of two similar labels on two similar jars of coffee is 144 : 169. Find the ratio of (a) the heights of the two jars (b) their capacities.
Working
(a) A1A2=left(h1h2right)2 ⇒ sqrt{144169=h1h2=1213
Result
(a) h1:h2=12:13
Working
(b) V1V2=left(h1h2right)3=left(1213right)3=17282197
Result
(b) V1:V2=1728:2197
R9.9.A tessellation of tiles on a floor has been made using a repeating pattern of a regular hexagon, six squares and six equilateral triangles. Find the total area of a single pattern with side length 1/2 metre of each polygon.
Given
The pattern forms a 12-sided regular polygon (dodecagon)
Formula
Area of dodecagon=3(2+sqrt3)a2
Working
=3(2+sqrt3) × left(frac12right)2=3(2+sqrt3) × frac14
Result
=2.8 m2