Question

The perimeter of a triangle is 42 cm the length og the smalledt side is 12
cm and the length of the longest side is 16 cm. find the length of the third side . find the area of the triangle
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Answer 1

Step-by-step explanation:

Correct Question:-

The perimeter of Triangle is 42 cm. The smallest side is 12 cm and the longest side is 16 cm. Find the area of the Triangle and the third Side.

Answer:-

We need to find the third side in order to find the area.

Concept:-

Heron's Formula and Perimeter.

Let's Do!

$12 + 16 + x = 42$

$28 + x = 42$

$x = 42 - 28$

$x = 14 \: cm$

Now,

$s = \dfrac{a + b + c}{2}$

$= \dfrac{12 + 16 + 14}{2}$

$= \dfrac{42}{2} = 21$

Now,

$area = \sqrt{s(s - a)(s - b)(s - c)}$

$\sqrt{21(21 -12)(21 - 16)(21 - 14) }$

$\sqrt{21 \times 9 \times 5 \times 7}$

$= \sqrt{6615}$

$= 81.33 \: {cm}^{2}$

Answer 2

Answer :-

$\: \: \: \underline{\boxed{\sf{Let's \: understand \: the \: concept \: first\: }}}$

Here the concept of Perimeter of Triangles and Area of Triangles whose all the sides are unequal. According to this, the perimeter of any triangle is equal to the sum of its all sides. And area we can find out using the semi perimeter.

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★ Formula to be used :-

$\: \: ➫ \: \: \boxed{\bold{Perimeter \: of \: Triangle \: = \: Sum \: of \: all \: the \: sides\:}}$

$\: \: ➫ \: \: \boxed{\bold{Area\: of \: a \: Triangle \: = \: \sqrt{s(s\: - \:a)(s\: - \:b)(s \: - \:c)}}}$

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★ Question :-

The perimeter of Triangle is 42 cm. The smallest side is 12 cm and the longest side is 16 cm. Find the third side and the area of triangle.

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★ Solution :-

Given,

» Smallest side of triangle = a = 12 cm

» Longest side of triangle = b = 16 cm

» Perimeter of Triangle = 42 cm

• First let us find the third side.

Let the length of the third side be 'c' cm.

Then,

➠ Perimeter of Triangle = Sum of All sides

➠ a + b + c = Perimeter of ∆

➠ 12 + 16 + c = 42

➠ 28 + c = 42

➠ c = 42 - 28

➠ c = 14 cm

$\: \: \boxed{\rm{Hence \: the \: length \: of \: the \: third \: side \: is \: \underline{14 \: cm}}}$

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• Now let us find the area. We already know the value of a, b and c.

So we need to find the value of semi perimeter(s) first.

This is given by,

$\: \: \Rightarrow \: \: \rm{s \: = \: \dfrac{a \: + \: b \: + \: c}{2}}$

$\: \: \Rightarrow \: \: \rm{s \: = \: \dfrac{42}{2}}$

⟹ Semi-Perimeter (s) = 21 cm

Now, using the formula of area of triangle we get,

$\: \: \sf{Area \: = \: \sqrt{21(21 \: - \: 12)(21 \: - \: 16)(21 \: - \: 14)}}$

➥ Area of Triangle = √(21 × 9 × 5 × 7)

➥ Area of Triangle = √(6615)

➥ Area of Triangle = 81.33 cm²

$\: \: ➮ \: \: \boxed{\rm{Hence \: the \: are \: of \: the \: triangle \: is \: \underline{81.33 \: cm²}}}$

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★ More to know :-

• Square is a closed figure with four sides perpendicular to each other and diagonals bisect each other.

• Rectangle is a closed figure where opposite sides are parallel to each other and all the sides are perpendicular to each other.

• Triangle is a closed figure with three sides may or may not be equal to each other.

• Circle is a closed figure where there is no end or starting point.