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 cm, and the parallelogran stands on the base 28 cm, find the height of the parallelogram.
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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 .
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 :–
Answer :–
we know that,
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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Answer :
• Height of the parallelogram = 12 cm
Given :
To find :
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,
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,
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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.