Question

If the sides of A triangle and a parallelogram have the same base and the the triangle are 26 cm, 28 cm and 30 cm, and the parallelogram stands on the base 28 cm, find the height of the parallelogram.

Answer 1

Answer:

Height of the parallelogram is 12 cm.

Step-by-step explanation:

Given: Area of the parallelogram = Area of the triangle

By using the area of the parallelogram formula, we can calculate the height of the parallelogram  

By using Heron’s formula, we can calculate the area of a triangle.

Heron's formula for the area of a triangle is: Area = √s(s - a)(s - b)(s - c)

Where a, b, and c are the sides of the triangle, and s = Semi-perimeter = Half the Perimeter of the triangle

Let ABCD is a parallelogram and ABE is a triangle having a common base with parallelogram ABCD.

For ∆ABE, a = 30 cm, b = 26 cm, c = 28 cm

Semi Perimeter: (s) = Perimeter/2

s = (a + b + c)/2

= (30 + 26 + 28)/2

= 84/2

= 42 cm

By using Heron’s formula,

Area of a ΔABE = √s(s - a)(s - b)(s - c)

= √42(42 - 30)(42 - 28)(42 - 26)

= √42 × 12 × 14 × 16

= 336 cm2

Area of parallelogram ABCD = Area of ΔABE (given)

Base × Height = 336 cm2

28 cm × Height = 336 cm2

On rearranging, we get

Height = 336/28 cm = 12 cm

Thus, height of the parallelogram is 12 cm.

Answer 2

⇝Given :-

Sides of a triangle have same base.

Sides of triangle = 26cm , 28cm , 30cm

Base of parallelogram = 28cm

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⇝To Find :-

Height of parallelogram = ?

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⇝Solution :-

❒ We know that :

${\large{\red{\bigstar \: \: \: \: \: \: {\orange{\underbrace{\underline{\green{\bf{Semi - Perimeter = \frac{a + b + c}{2} }}}}}}}}}$

${\large{\red{\bigstar \: \: \: \: \: \: {\orange{\underbrace{\underline{\green{\bf{Area = \sqrt{s(s - a)(s - b)(s - c)} }}}}}}}}}$

❒ Area of triangle :

✏ Semi - Perimeter :

${\large{:{\longmapsto{\bf{Semi - Perimeter = \frac{a + b + c}{2} }}}} }$

${\large{:{\longmapsto{\bf{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \frac{26 + 28 + 30}{2} }}}} }$

${\large{:{\longmapsto{\bf{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: {\cancel \frac{84}{2} }}}} }}$

${\large{\orange{\dashrightarrow{\blue{\underline{\bf{42 cm}}}}}}}$

✏ Area :

${\large{:{\longmapsto{\bf{Area = \sqrt{s(s - a)(s - b)(s - c)} }}}} }$

${\large{:{\longmapsto{\bf{Area = \sqrt{42(42 - 26)(42 - 28)(42 - 30)} }}}} }$

${\large{:{\longmapsto{\bf{Area = \sqrt{42 \times 16 \times 14 \times 12} }}}} }$

${\large{\orange{\dashrightarrow{\blue{\underline{\bf{336 \: {cm}^{2} }}}}}}}$

❒ Height of parallelogram :

✏ Here :

Base = 28 cm

Area = area of triangle = 336 cm²(given)

Height = ?

✏ Height :

${\large{:{\longmapsto{\bf{Area \: of \: parallelogram = Base \times Height}}}}}$

${\large{:{\longmapsto{\bf{336 = 28 \times height}}}}}$

${\large{:{\longmapsto{\bf{Height = {\cancel\frac{336}{28} }}}}}}$

${\large{\red{:{\twoheadrightarrow{\purple{\underline{\overline{\boxed{\bf{Height = 12 cm}}}}}}}}}}$

❒ Hence :

${\huge{\purple{\underline{\red{\underline{\pink{\pmb{\mathfrak{Height = 12cm}}}}}}}}}$

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