Question

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

Answer 1

Question

A triangle and a parallelogram have the same base and the same area. If the sides of the triangle are 26 cm. 28 cm ana 30 cm, and the parallelogram stands on the base28 cm, find the height of the parallelogram

Solution

$\underline{\green{\sf{Given\ That}}}$

• Triangle and a Parallelogram have the same base and the same area
• Side of triangle are 26 cm, 28 cm and 30 cm
• Base of parallelogram is 28 cm

$\underline{\green{\sf{To ~ Find}}}$

• Height of parallelogram

$\underline{\green{\sf{Step\ by\ step\ explanation}}}$

Now in triangle ABC we have

$\sf{Area\ of\ triangle\ = \sqrt{s(s-a)(s-b)(s-c)}}$

$\text{Where }$

$\text{S is Semi- perimeter}$

$\text{a , b and c are the sides of triangle}$

$\sf{Semi-Perimeter = \dfrac{ Sum\ of \ all \ sides }{2}}$

$\sf{Semi-Perimeter = \dfrac{ 26cm+28+30 }{2}}$

$\sf{Semi-Perimeter = \dfrac{84cm}{2}}$

$\sf{Semi-Perimeter = 42cm}$

$\\ { \sf{Area\ of\ triangle\ \implies \sqrt{s(s-a)(s-b)(s-c)}}}\\ \\ { \implies \sf \sqrt{42(42-26)(42-28)(42-30)}} \\ \\{\implies\sf \sqrt{42(16)(14)(12)}} \\ \\ {\implies\sf\sqrt{42×2688}} \\ \\ \: {\implies\sf\sqrt{112896}} \\ \\ \: \sf {\implies\sqrt{336²}} \\ \\ {\implies\sf 336cm²}$

$\text{Now it is given That }$

• Triangle and a parallelogram have the same base and the same area

$\sf{Area\ of\ parallelogram\ = Area\ of \ triangle}\\\\\sf{Length×Height = 336}\\\\\sf{28×Height =336}\\\\\sf{Height =\dfrac{336}{28}}\\\\\sf{Height =12cm}$

$\text{The height of the parallelogram is \pink{12cm}.}$

Answer 2

Given:

A triangle and a parallelogram with

• Same Base
• Same Area

A triangle with

• 1st side = 26 cm
• 2nd side = 28 cm
• 3rd side = 30 cm

A parallelogram with

• Base = 28 cm

What To Find:

We have to find

• The height of the parallelogram.

How To Find:

To find the height, we have to,

• First, find the area of the triangle using a certain formula.
• Next, find the height of the parallelogram using a certain formula.

Formulas Needed:

• For the area of a triangle -

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

Where,

• s = Semi-perimeter
• a = 1st side
• b = 2nd side
• c = 3rd side

• For the area of a parallelogram -

$\bf \longmapsto Area = b \times h$

Where,

• b = Base
• h = Height

Solution:

• Finding the area of the triangle.

Finding the semi-perimeter.

⇒ Perimeter = Sum of three sides.

⇒ Perimeter = 26 + 28 + 30

⇒ Perimeter = 84 cm

⇒ Semi-perimeter = Perimeter ÷ 2

⇒ Semi-perimeter = 84 ÷ 2

⇒ Semi-perimeter = 42 cm

Finding the area.

$\sf \implies Area = \sqrt{s(s-a)(s-b)(s-c)}$

$\sf \implies Area = \sqrt{42(42-26)(42-28)(42-30)}$

$\sf \implies Area = \sqrt{42(16)(14)(12)}$

$\sf \implies Area = \sqrt{112896}$

$\sf \implies Area = 336 \: cm^2$

• Finding the height of the parallelogram.

⇒ Area of the triangle = Area of the parallelogram = 336 cm²

⇒ Area = b × h

⇒ 336 cm² = 28 × h

⇒ 336 ÷ 28 = h

⇒ 12 cm = h

Final Answer:

∴ Thus, the height of the parallelogram is 12 cm.