Question

5. Using Heron's formule, find the area of an isosceles triangle whose perimeter is 16cm and base is 6 cm.
with solution ​

Answer 1

Given: The area of an isosceles triangle whose Perimeter is 16 cm. & The Base of the isosceles triangle is 6 cm.

Need to find: Area of the triangle?

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» Let's say, the equal sides of the isosceles triangle be x cm.

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$\underline{\bf{\dag} \:\mathfrak{As\;we\;know\: that\: :}}$⠀⠀⠀⠀

• Perimeter of the triangle is sum of all sides. & Perimeter is Given that is 16 cm.

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$:\implies\bf \Big\{Perimeter = x + x + c\Big\}\\\\\\:\implies\sf 2x + 6 = 16\\\\\\:\implies\sf 2x = 16 - 6\\\\\\:\implies\sf 2x = 10\\\\\\:\implies\sf x = \cancel\dfrac{10}{2}\\\\\\:\implies{\pmb{\boxed{\frak{x = 5}}}}$

∴ Hence, the sides of the isosceles triangle are 5 cm, 5 cm & 6 cm respectively.

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✇ & If the perimeter of the Given Isosceles triangle is 16 cm then the semi – perimeter of the triangle would be 8 cm. i.e ( s ) = 8 cm.

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• C a l c u l a t i n G⠀A r e a :

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$\star\;{\underline{\boxed{\pmb{\sf{\mathfrak{A}rea_{\:(triangle)} = \sqrt{s\Big(s - a\Big) \Big(s - b\Big) \Big(s - c\Big)}}}}}}$

$\frak{Sides}\begin{cases}\sf{\quad a =\bf{5\;cm}}\\\sf{\quad b =\bf{5\;cm}}\\\sf{\quad c=\bf{6\;cm}}\\\sf{\quad s = \bf{8\:cm}}\end{cases}$

$\dashrightarrow\sf Area_{\:(triangle)} = \sqrt{8\Big(8 - 5\Big)\Big(8 - 5\Big)\Big(8 - 6\Big)}\\\\\\\dashrightarrow\sf Area_{\;(triangle)} = \sqrt{8 \times 3 \times 3 \times 2}\\\\\\\dashrightarrow\sf Area_{\:(triangle)} = \sqrt{144}\\\\\\\dashrightarrow{\pmb{\boxed{\frak{Area_{\:(triangle)} = 12\;cm^2}}}}\;\bigstar$

$\therefore{\underline{\sf{Hence,\;the\;area\;of\; isosceles\; triangle\;is\;{\pmb{\sf{12\;cm^2}}}.}}}$⠀⠀⠀⠀⠀⠀

Answer 2

Question:

• Using Heron's formula, find the area of an isosceles triangle whose perimeter is 16 cm and base is 6 cm.

Answer:

• Area of an isosceles triangle is 12 cm².

Explanation:

Given that:

• Perimeter of ∆ = 16 cm
• Base of ∆ = 6 cm

To Find:

• Area of ∆?

Solution:

• Firstly let's find unknown sides that are equal because ∆ is isosceles.
• Let the equal sides be n cm.

We know that,

✪ Perimeter of ∆ = Sum of all sides ✪

According to the question putting all values in formula we get,

➻ 16 = n + n + 6

➻ 2n + 6 = 16

➻ 2n = 16 - 6

➻ 2n = 10

➻ n = $\sf {\cancel{\dfrac{10}{5}}}$

➻ $\bf \red{n = 5}$

∴ Unknown sides of ∆ are 5 cm and 5 cm.

• Now, finding semi perimeter (s) of ∆ ::

We know that,

✪ $\bf s = \dfrac{Perimeter}{2}$ ✪

According to the question putting all values in formula we get,

➻ $\sf s = {\cancel{\dfrac{16}{2}}}$

➻ $\bf \purple{s = 8}$

∴ Semi perimeter (s) of ∆ is 8 cm.

• Now, we have all required values. So, let's find area of ∆ ::

According to heron's formula we know that,

✪ Area of ∆ = √[s(s - a)(s - b)(s - c)] ✪

• Where, s is semi perimeter and a, b, c are sides of ∆. We have, s = 8 cm, a = 5 cm, b = 5 cm and c = 6 cm.

According to the question putting all values in formula we get,

➻ Area of ∆ = √[8(8 - 5)(8 - 5)(8 - 6)]

➻ Area of ∆ = √(8 × 3 × 3 × 2)

➻ Area of ∆ = √144

➻ Area of ∆ = √(12 × 12)

➻ $\bf\pink{Area\:of\:\triangle = 12\:cm^2}$

∴ Area of an isosceles triangle is 12 cm².

Know more:

• Area of rectangle = 2(L + B)
• Area of square = (side)²
• Area of circle = πr²
• Area of equilateral ∆ = √3/4 × (side)²
• Area of trapezium = ½ × (a + b) × h

Learn more on branly:

✧ Related Question ✧

• An isosceles triangle has perimeter 30 cm and each of the equal sides is 12 cm. Find the area of the triangle.

✧ Answer ✧

• brainly.in/question/43884844

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