Question

answer with solution please​

Answer 1

$\\\;\underbrace{\underline{\sf{Understanding\;the\;Concept}}}$

Here the concept of Properties of Isosceles Triangle and Heron's Formula has been used. We know that two sides of the isoceles triangle excluding the base are equal in length. Also we are given the length of base. Using base and perimeter we can find the length of those sides. After that we can find the area of triangle using Heron's Formula.

Let's do it !!

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★ Formula Used :-

$\\\;\boxed{\sf{\pink{Perimeter\;of\;\Delta\;=\;\bf{Sum\;of\;all\;sides}}}}$

$\\\;\boxed{\sf{\pink{Semi\:-\:Perimeter\;of\;\Delta,\;s\;=\;\bf{\dfrac{Perimeter\;of\;\Delta}{2}}}}}$

$\\\;\boxed{\sf{\pink{Area\;of\;\Delta\;=\;\bf{\sqrt{s(s\:-\:a)(s\:-\:b)(s\:-\:c)}}}}}$

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★ Solution :-

Given,

» Base of the triangle = 12 cm

» Perimeter of the triangle = 32 cm

• Let each of the two equal sides of the triangle be 'x'

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~ For the Length of Equal sides of ∆ ::

We know that,

$\\\;\sf{\rightarrow\;\;Perimeter\;of\;\Delta\;=\;\bf{Sum\;of\;all\;sides}}$

Now by applying values, we get

$\\\;\sf{\rightarrow\;\;32\;=\;\bf{x\;+\;x\;+\;12}}$

$\\\;\sf{\rightarrow\;\;32\;=\;\bf{2x\;+\;12}}$

$\\\;\sf{\rightarrow\;\;2x\;=\;\bf{32\;-\;12}}$

$\\\;\sf{\rightarrow\;\;2x\;=\;\bf{20}}$

$\\\;\sf{\rightarrow\;\;x\;=\;\bf{\dfrac{20}{2}}}$

$\\\;\bf{\rightarrow\;\;x\;=\;\bf{\green{10\;\;cm}}}$

Hence, two opposite sides of triangle are of 10 cm.

So length of all the sides of triangle = 10 cm, 10 cm and 12 cm

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~ For the Semi - Perimeter of the ∆ ::

s denotes the Semi - Perimeter of triangle.

We know that,

$\\\;\sf{\rightarrow\;\;Semi\:-\:Perimeter\;of\;\Delta,\;s\;=\;\bf{\dfrac{Perimeter\;of\;\Delta}{2}}}$

By applying values, we get

$\\\;\sf{\rightarrow\;\;Semi\:-\:Perimeter\;of\;\Delta,\;s\;=\;\bf{\dfrac{32}{2}}}$

$\\\;\bf{\rightarrow\;\;Semi\:-\:Perimeter\;of\;\Delta,\;s\;=\;\bf{\red{16\;\;cm}}}$

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~ For the Area of the Triangle ::

We found that, all the sides of triangle :-

• Base = a = 12 cm

• One of equal side = b = 10 cm

• Other equal side = c = 10 cm

• Semi - Perimeter of triangle = s = 16 cm

According to heron's formula, we know that

$\\\;\sf{\Longrightarrow\;\;Area\;of\;\Delta\;=\;\bf{\sqrt{s(s\:-\:a)(s\:-\:b)(s\:-\:c)}}}$

By applying values, we get

$\\\;\sf{\Longrightarrow\;\;Area\;of\;\Delta\;=\;\bf{\sqrt{(16)(16\:-\:12)(16\:-\:10)(16\:-\:10)}}}$

$\\\;\sf{\Longrightarrow\;\;Area\;of\;\Delta\;=\;\bf{\sqrt{(16)(4)(6)(6)}}}$

$\\\;\sf{\Longrightarrow\;\;Area\;of\;\Delta\;=\;\bf{\sqrt{2304}}}$

We know that, 48 × 48 = 48² = 2304. So,

$\\\;\sf{\Longrightarrow\;\;Area\;of\;\Delta\;=\;\bf{\sqrt{48\:\times\:48}}}$

$\\\;\sf{\Longrightarrow\;\;Area\;of\;\Delta\;=\;\bf{\sqrt{48^{2}}}}$

$\\\;\bf{\Longrightarrow\;\;Area\;of\;\Delta\;=\;\bf{\orange{48\;\:cm^{2}}}}$

$\\\;\underline{\boxed{\tt{Hence,\;\:area\;\:of\;\:\Delta\;=\;\bf{\purple{48\;\;cm^{2}}}}}}$

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★ More to know :-

$\\\;\sf{\leadsto\;\;Area\;of\;Triangle\;=\;\dfrac{1}{2}\:\times\:Base\:\times\:Height}$

$\\\;\sf{\leadsto\;\;Area\;of\;Parallelogram\;=\;2\:\times\:Area\;of\;\Delta}$

$\\\;\sf{\leadsto\;\;Area\;of\;Equilateral\;\Delta\;=\;\dfrac{\sqrt{3}}{4}\:a^{2}}$

• A triangle is formed by joining three sides end to end.

• Sum of all angles of triangle sum to 180° . This is also known as Angle Sum Property of triangle.

• On the basis of length of sides, a triangle is divided into Equilateral, Isoceles and Scalene.

• On the basis of angle, a triangle is divided into Acute, Obtuse and Right - Angled Triangle.

Answer 2

Answer☞

Let the congruent sides of the isosceles traingle=a and the base=b.

Given,b=12cm

Perimeter of the traingle=Sum of all sides=a+a+b=32

=>2a+12=32

On solving,

a=10 cm

Since the height of the traingle,divides it into two right angled traingles.

a²=h²+(b/2)²=10²=h²+6²

h²=100-36=64

h=√64=8 cm

Area of traingle=½×base×height

½×12×8

48 sq cm