NCERT Class 9 Solutions: Heron's Formula (Chapter 12) Exercise 12.2 Part 2

Q-3 Radha made a picture of an airplane with colored paper as shown in figure. Find the total area of the paper used.

Triangle poly1 Triangle poly1: Polygon A, B, C Quadrilateral poly2 Quadrilateral poly2: Polygon A, D, E, C Triangle poly3 Triangle poly3: Polygon C, F, G Triangle poly4 Triangle poly4: Polygon A, H, I Quadrilateral poly5 Quadrilateral poly5: Polygon D, J, K, E Segment c Segment c: Segment [A, B] of Triangle poly1 Segment a Segment a: Segment [B, C] of Triangle poly1 Segment b Segment b: Segment [C, A] of Triangle poly1 Segment a_1 Segment a_1: Segment [A, D] of Quadrilateral poly2 Segment d Segment d: Segment [D, E] of Quadrilateral poly2 Segment e Segment e: Segment [E, C] of Quadrilateral poly2 Segment c_1 Segment c_1: Segment [C, A] of Quadrilateral poly2 Segment g Segment g: Segment [C, F] of Triangle poly3 Segment c_2 Segment c_2: Segment [F, G] of Triangle poly3 Segment f Segment f: Segment [G, C] of Triangle poly3 Segment i Segment i: Segment [A, H] of Triangle poly4 Segment a_2 Segment a_2: Segment [H, I] of Triangle poly4 Segment h Segment h: Segment [I, A] of Triangle poly4 Segment d_1 Segment d_1: Segment [D, J] of Quadrilateral poly5 Segment j Segment j: Segment [J, K] of Quadrilateral poly5 Segment k Segment k: Segment [K, E] of Quadrilateral poly5 Segment e_1 Segment e_1: Segment [E, D] of Quadrilateral poly5 Vector u Vector u: Vector[L, M] Vector u Vector u: Vector[L, M] Vector v Vector v: Vector[N, O] Vector v Vector v: Vector[N, O] Vector w Vector w: Vector[P, Q] Vector w Vector w: Vector[P, Q] Vector l Vector l: Vector[R, S] Vector l Vector l: Vector[R, S] Vector m Vector m: Vector[T, U] Vector m Vector m: Vector[T, U] Vector n Vector n: Vector[V, W] Vector n Vector n: Vector[V, W] Vector p Vector p: Vector[Z, A_1] Vector p Vector p: Vector[Z, A_1] Vector q Vector q: Vector[B_1, C_1] Vector q Vector q: Vector[B_1, C_1] 5cm text1 = "5cm" 6cm text2 = "6cm" 1.5cm text3 = "1.5cm" 6.5cm text4 = "6.5cm" 1cm text5 = "1cm" 1cm text6 = "1cm" 1cm text7 = "1cm" 2cm text8 = "2cm" i text9 = "i" iii text10 = "iii" v text11 = "v" v text12 = "v" ii text13 = "ii" iv text14 = "iv"

Picture of an Airplane

Picture of an airplane made with colored paper

Solution:

We can divide the airplane into the separate quadrilaterals and triangles. Let’s find each of their areas

For triangle (i)

Triangle poly1 Triangle poly1: Polygon A, B, C Segment c Segment c: Segment [A, B] of Triangle poly1 Segment a Segment a: Segment [B, C] of Triangle poly1 Segment b Segment b: Segment [C, A] of Triangle poly1 Vector u Vector u: Vector[D, E] Vector u Vector u: Vector[D, E] Vector v Vector v: Vector[F, G] Vector v Vector v: Vector[F, G] Vector w Vector w: Vector[H, I] Vector w Vector w: Vector[H, I] Vector d Vector d: Vector[J, K] Vector d Vector d: Vector[J, K] 5cm text1 = "5cm" 1cm text2 = "1cm"

Isolated Top Triangle From the Airplane.

Isolated triangle from the airplane.

This triangle is an isosceles triangle, therefore its perimeter =(5+5+1)cm=11cm and semi-perimeter s=11cm2=5.5cm

By using heroes formula, area of the triangle

  • =s(sa)(sb)(sc)

  • =5.5(5.55)(5.55)(5.51)cm2

  • =(5.5)(0.5)(0.5)(4.5)cm2

  • =6.1875

  • =0.7511cm2

  • =0.75×3.317cm2

  • =2.488cm2 (approx.)

For quadrilateral (ii)

Quadrilateral poly1 Quadrilateral poly1: Polygon A, B, C, D Segment a Segment a: Segment [A, B] of Quadrilateral poly1 Segment b Segment b: Segment [B, C] of Quadrilateral poly1 Segment c Segment c: Segment [C, D] of Quadrilateral poly1 Segment d Segment d: Segment [D, A] of Quadrilateral poly1 Vector u Vector u: Vector[E, F] Vector u Vector u: Vector[E, F] Vector v Vector v: Vector[G, H] Vector v Vector v: Vector[G, H] 6.5cm text1 = "6.5cm" 1cm text2 = "1cm"

Rectangle in Middle Part of Aircraft

Rectangle thatlength=6.5cmandbase=1cm

Area of Rectangle =l×b =(6.5×1)cm2=6.5cm2

For quadrilateral (iii)

Quadrilateral poly1 Quadrilateral poly1: Polygon A, B, C, D Segment a Segment a: Segment [A, B] of Quadrilateral poly1 Segment b Segment b: Segment [B, C] of Quadrilateral poly1 Segment c Segment c: Segment [C, D] of Quadrilateral poly1 Segment d Segment d: Segment [D, A] of Quadrilateral poly1 1cm text1 = "1cm" 2cm text2 = "2cm" 1cm text3 = "1cm" 1cm text4 = "1cm"

Trapezium at Tail of Aircraft

Trapezium at tail of aircraft

This quadrilateral is a trapezium. To find the area we need to first find its height. = (12(0.5)2)cm

  • Perpendicular height of parallelogram

    • (0.75)cm

    • 0.866cm

  • Area=area of parallelogram+ area of equilateral triangle

    • b×h+34a2

    • (0.866)×1+34(1)2

    • (0.866)+0.433=1.299cm2

For triangle (IV) = triangle (v)

We know areaoftriangle=12h×b

  • 12×1.5cm×6cm

  • 2×4.5cm2 (2 triangle)

  • 9cm2

Therefore, total area of the paper used

  • (2.488+6.5+1.299+9)cm2

  • 19.28cm2

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