The ratio of the measures of three sides of
a triangle is 4:2:3 and its perimeter is
36 cm. Find the area of this triang

2 thoughts on “The ratio of the measures of three sides of<br />a triangle is 4:2:3 and its perimeter is<br />36 cm. Find the area of this triang”

1. Answer:

   taking sides of triangle in ratio of 4x, 2x, 3x and perimeter is 36 cm.

   Therefore, perimeter of triangle = Sum of all sides

   36 = 4x + 2x + 3x

   36 = 9x

   4 = x

   x = 4

   Therefore , sides of triangle are

   4x = 4×4 = 16 cm

   2x = 2×4 = 8 cm

   3x = 3×4 = 12 cm

   Now, using heron’s formula

   s = (a+b+c)/2

   s = (16+8+12)/2

   s = 36/2

   s = 18

   Now, Area = √s(s-a)(s-b)(s-c)

   Area = √18(18-16)(18-8)(18-12)

   Area = √18×2×10×6

   Area = √2160

   Area = 46.47 cubic cm.

   Area of a triangle is 46.47 cubic cm.

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2. Let each side of the triangle be 4x, 2x and 3x

   [tex]\\[/tex]

   Perimeter = Sum of all sides

   → 36 = 4x + 2x + 3x

   → 36 = 9x

   → x = 36 ÷ 9

   → x = 4 cm

   [tex]\\[/tex]

   For finding each side of the triangle,

   • 4x = 4(4) = 16 cm = a
   • 2x = 2(4) = 8 cm = b
   • 3x = 3(4) = 12 cm = c

   [tex]\\[/tex]

   Since we know the 3 sides of the triangle, we apply Herons formula to find the area of triangle.

   [tex]\\[/tex]

   Semiperimeter = 36 ÷ 2 = 18

   So, s = 18

   [tex]\\[/tex]

   [tex] \sf{Area = \sqrt{s(s – a)(s – b)(s – c)} } \\ \\ [/tex]

   [tex]\sf{Area = \sqrt{18(18 – 16)(18 – 8)(18 – 12)} } \\ \\ [/tex]

   [tex] : \implies \sf{Area = \sqrt{18(2)(10)(6)} } \\ \\ [/tex]

   [tex]: \implies \sf{Area = \sqrt{36 \times 60} } \\ \\ [/tex]

   [tex]: \implies \sf{Area = 6 \sqrt{4 \times 15} } \\ \\ [/tex]

   [tex]: \implies \sf{Area = 12 \sqrt{15} } \\ [/tex]

   Substitute the value of √15 as 3.87

   [tex] \\ \therefore \boxed{ \bf{Area = 46.44\: {cm}^{2} }}[/tex]

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