NCERT Class 9 Solutions: Heron's Formula (Chapter 12) Exercise 12.1 Part 1

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Herons formula with semi-perimeter

Herens Formula in Two Parts

Herons formula with semi-perimeter

Q-1 A traffic signal board, indicating 'SCHOOL AHEAD', is an equilateral triangle with side 'a'. Find the area of the signal board, using Heron’s formula. If its perimeter is 180 cm, what will be the area of the signal board?

Solution:

  • Length of equilateral triangle =a

  • Perimeter of the signal board =3a=180cm3a=180cma=60cm

  • Semi perimeter of the signal board (s) =3a2 (s=perimeter2)

  • Using heron's formula

    Area of the signal board

    • s(sa)(sb)(sc)

    • (3a2)(3a2a)(3a2a)(3a2a)

    • 3a2×a2×a2×a2

    • 3a416

    • 3a24

    • 34×60cm×60cm=9003cm2(a=60cm)

Q-2 The triangular side walls of a flyover have been used for advertisements. The sides of the walls are 122m,22mand120m (see Fig). The advertisements yield on earning of 5000perm2 per year. A company hired one of its walls for 3 months. How much rent did it pay?

Triangular side walls used for advertisement, side of wall are 122m,22m and 120m

Triangular Side Walls Used for Advertisement

Triangular side walls used for advertisement, side of wall are 122m,22m and 120m

Solution:

Given, the sides of the triangle are 122m,22mand120m.

  • Perimeter of the triangle is 122+22+120=264m ( Perimeter=a+b+c )

  • Semi perimeter of triangle (s) =2642=132m ( semiperimeter=a+b+c2 )

  • Using heron's formula,

    Area of the advertisement

    • s(sa)(sb)(sc)

    • 132(132122)(13222)(132120)m2

    • 132×10×110×12m2

    • 1742400m2

    • 1320m2

Rate of advertising rent per year = 5000perm2

Therefore, Rent of one wall for 3 months = 1320×5000×312=1650000

Q-3 There is a slide in a park. One of its side walls has been painted in some color with a message “KEEP THE PARK GREEN AND CLEAN” (see Fig). If the sides of the wall are 15m,11mand6m, finding the area painted in colour.

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 Segment f Segment f: Segment [C, D] Segment g Segment g: Segment [A, E] Segment h Segment h: Segment [D, E] Segment i Segment i: Segment [B, F] Segment j Segment j: Segment [E, F] Vector u Vector u: Vector[G, H] Vector u Vector u: Vector[G, H] Vector v Vector v: Vector[I, J] Vector v Vector v: Vector[I, J] Vector w Vector w: Vector[K, L] Vector w Vector w: Vector[K, L] Vector d Vector d: Vector[M, N] Vector d Vector d: Vector[M, N] 11m text1 = "11m" 15m text2 = "15m" 6m text3 = "6m" KEEP THE PARK GARDEN AND CLEAN text4 = "KEEP THE PARK GARDEN AND CLEAN" KEEP THE PARK GARDEN AND CLEAN text4 = "KEEP THE PARK GARDEN AND CLEAN"

Slide in a Park

Slide in a park, its side painted in some message “KEEP THE PARK GREEN AND CLEAN”.

Solution:

  • Sides of the triangular wall are 15m,11mand6m.

  • Semi perimeter of triangular wall (s) =15+11+62m=16m

  • Using heron's formula, area of the message

    • s(sa)(sb)(sc)

    • 16(1615)(1611)(166)m2

    • 16×1×5×10m2

    • (800m2)

    • 202m2

Q-4 Find the area of a triangle two sides of which are 18cmand10cm and the perimeter is 42cm.

Solution:

  • Two sides of the triangle =18cmand10cm

  • Perimeter of the triangle =42cm

  • Third side of triangle =42(18+10)cm=14cm ( perameter=a+b+c=42=18+10+cc(thirdside)=421810 )

  • Semi perimeter of triangle =422=21cm (s=a+b+c2)

  • Using heron's formula,

    Area of the triangle

    • s(sa)(sb)(sc)

    • 21(2118)(2110)(2114)cm2

    • 21×3×11×7m2

    • 4851

    • 2111cm2

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