4. A triangle and a parallelogram have the same base and the same area. If the sides of
the triangle are 26 cm, 28 cm and 30

2 thoughts on “4. A triangle and a parallelogram have the same base and the same area. If the sides of<br />the triangle are 26 cm, 28 cm and 30”

1. Given :– A triangle and a parallelogram have the same base and the same area. If the sides of the triangle are 26 cm, 28 cm and 30 cm, and the parallelogran stands on the base 28 cm .

   To Find :–

   • The height of the parallelogram. ?

   Answer :–

   we know that,

   • Area of triangle = √[s * (s – a) * (s – b) * (s – c)] where s is semi – perimeter and a , b and c are side lengths .
   • Area of parallelogram = Base * height .

   so,

   → semi – perimeter of ∆ = (a + b + c)/2 = (26 + 28 + 30)/2 = 84/2 = 42 cm .

   Now, Let height of parallelogram is h cm .

   then, comparing both area we get,

   → Area of parallelogram = Area of triangle

   → 28 * h = √[42 * (42 – 26) * (42 – 28) * (42 – 30)]

   → 28 * h = √[ 42 * 16 * 14 * 12]

   → 28 * h = √[7 * 6 * 4 * 4 * 7 * 2 * 6 * 2]

   → 28 * h = √(7² * 6² * 4² * 2²)

   → 28 * h = 7 * 6 * 4 * 2

   → 28 * h = 28 * 12

   → h = 12 cm (Ans.)

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2. Answer :

   • Height of the parallelogram = 12 cm

   Given :

   • Base of triangle = Base of parallelogram
   • Area of triangle = Area of parallelogram
   • The three sides of the triangle as 26 cm, 28 cm and 30 cm respectively.

   To find :

   • Height of the parallelogram

   Concept :

   Formula to calculate Semi – perimeter :-

   [tex]\boxed{\bold{Semi – perimeter = \dfrac{a + b + c}{2}}}[/tex]

   where,

   • a denotes the first side of the triangle

   • b denotes the second side of the triangle

   • c denotes the third side of the triangle

   Formula to calculate Area of triangle, when the three sides of triangle are given or Heron’s formula :-

   [tex]\boxed{\bold{Heron’s~formula = \sqrt{s(s – a)(s – b)(s – c)}}}[/tex]

   where,

   • s denotes the semi – perimeter

   Formula to calculate Area of parallelogram :

   [tex]\boxed{\bold{Area~of~parallelogram = b \times h}}[/tex]

   where,

   • b denotes the base of parallelogram

   • h denotes the height of parallelogram

   Solution :

   To calculate the height of the parallelogram, firstly calculate the area of the triangle as the area of triangle is equal to area of parallelogram.

   Semi – perimeter of the triangle :

   [tex]\\ \twoheadrightarrow \quad\sf{Semi – perimeter = \dfrac{a + b + c}{2}}[/tex]

   Let us assume,

   • a = 26 cm
   • b = 28 cm
   • c = 30 cm

   Substituting the given values :

   [tex]\\ \twoheadrightarrow \quad\sf{Semi – perimeter = \dfrac{26 + 28 + 30}{2}}[/tex]

   [tex]\\ \twoheadrightarrow \quad\sf{Semi – perimeter = \dfrac{84}{2}}[/tex]

   [tex]\\ \twoheadrightarrow \quad\sf{Semi – perimeter = \dfrac{ \not84}{ \not2}}[/tex]

   [tex]\\ \twoheadrightarrow \quad\sf{Semi – perimeter = 42}[/tex]

   Semi – perimeter of triangle = 42 cm

   Now, calculate the area of triangle by using the Heron’s formula :

   [tex]\\ \twoheadrightarrow \quad\sf{Heron’s~formula = \sqrt{s(s – a)(s – b)(s – c)}}[/tex]

   Substituting the given values :

   [tex]\\ \twoheadrightarrow \quad\sf{Area \: of \: triangle = \sqrt{42(42 – 26)(42- 28)(42 – 30)}}[/tex]

   [tex]\\ \twoheadrightarrow \quad\sf{Area \: of \: triangle = \sqrt{42(16)(14)(12)}}[/tex]

   [tex]\\ \twoheadrightarrow \quad\sf{Area \: of \: triangle = \sqrt{7 \times 6 \times 2 \times 2 \times 2 \times 2 \times 7 \times 2 \times 6 \times 2}}[/tex]

   [tex]\\ \twoheadrightarrow \quad\sf{Area \: of \: triangle = 7 \times 6 \times 2 \times 2 \times 2}[/tex]

   [tex]\\ \twoheadrightarrow \quad\sf{Area \: of \: triangle = 336}[/tex]

   Area of the triangle = 336 cm²

   Area of triangle = Area of parallelogram

   ∴ Area of parallelogram = 336 cm²

   Now, to calculate the height of the parallelogram, substitute the values in the formula of area of parallelogram :

   [tex] \\ \twoheadrightarrow \quad\sf Area~of~parallelogram = b \times h[/tex]

   [tex] \\ \twoheadrightarrow \quad\sf 336 = 28 \times h[/tex]

   Transposing 28 to the left hand side :

   [tex] \\ \twoheadrightarrow \quad\sf \dfrac{336}{28} = h[/tex]

   Dividing both the numbers by 2 :

   [tex] \\ \twoheadrightarrow \quad\sf \dfrac{168}{14} = h[/tex]

   Dividing both the numbers by 2 :

   [tex] \\ \twoheadrightarrow \quad\sf \dfrac{84}{7} = h[/tex]

   Dividing both the numbers by 7 :

   [tex] \\ \twoheadrightarrow \quad\sf 12 = h[/tex]

   Therefore,

   • Height of the parallelogram = 12 cm

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   VERIFICATION :

   To verify the value of height of the parallelogram, substitute the value of b and h in the expression “336 = b × h”.

   LHS = 336

   Taking RHS :

   ⠀⠀⠀⇒ b × h

   ⠀⠀⠀⇒ 28 × 12

   ⠀⠀⠀⇒ 336

   RHS = 336

   LHS = RHS

   HENCE, VERIFIED.

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