An isosceles triangle has perimeter $30cm$ and each of the equal sides is $12cm$. Find the area of the triangle.
Last updated date: 20th Mar 2023
•
Total views: 306.9k
•
Views today: 5.85k
Answer
306.9k+ views
Hint: Determine the sides of the triangle first. And then use Heron’s formula for the area of a triangle.
According to the question, the perimeter of the triangle is $30cm$. Then the semi-perimeter will be:
$ \Rightarrow s = \dfrac{{30}}{2} = 15cm$.
Length of two equal sides of the triangle is $12cm$ (given in the question).
Let the length of the third side is $x$. Then we can use perimeter to find the value of $x$. We’ll get:
$
\Rightarrow 12 + 12 + x = 30, \\
\Rightarrow x = 30 - 24, \\
\Rightarrow x = 6 \\
$
So, we have $12cm, 12cm$ and $6cm$ as the length of three sides of a triangle whose semi-perimeter is $s = 15cm$.
Now, we can use Heron’s Formula to determine its area. We have:
$ \Rightarrow A = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} $, where $a,b,c$ are length of sides of triangle in any order.
So, putting values from above, we’ll get:
$
\Rightarrow A = \sqrt {15\left( {15 - 12} \right)\left( {15 - 12} \right)\left( {15 - 6} \right)} , \\
\Rightarrow A = \sqrt {15 \times 3 \times 3 \times 9} , \\
\Rightarrow A = 3 \times 3 \times \sqrt {15} , \\
\Rightarrow A = 9\sqrt {15} \\
$
Thus the area of the triangle is $9\sqrt {15} c{m^2}$.
Note:
We can also use $Area = \dfrac{1}{2} \times b \times h$ to find out its area.
For an isosceles triangle, base is always the unequal side. So, in this case $b = 6cm$.
And we can easily find out the height of an isosceles using Pythagoras Theorem.
$
\Rightarrow h = \sqrt {{{12}^2} - {{\left( {\dfrac{b}{2}} \right)}^2}} , \\
\Rightarrow h = \sqrt {144 - 9} , \\
\Rightarrow h = \sqrt {135} , \\
\Rightarrow h = 3\sqrt {15} \\
$
Now, we can use $\dfrac{1}{2} \times b \times h$. We’ll get the same result.
According to the question, the perimeter of the triangle is $30cm$. Then the semi-perimeter will be:
$ \Rightarrow s = \dfrac{{30}}{2} = 15cm$.
Length of two equal sides of the triangle is $12cm$ (given in the question).
Let the length of the third side is $x$. Then we can use perimeter to find the value of $x$. We’ll get:
$
\Rightarrow 12 + 12 + x = 30, \\
\Rightarrow x = 30 - 24, \\
\Rightarrow x = 6 \\
$
So, we have $12cm, 12cm$ and $6cm$ as the length of three sides of a triangle whose semi-perimeter is $s = 15cm$.
Now, we can use Heron’s Formula to determine its area. We have:
$ \Rightarrow A = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} $, where $a,b,c$ are length of sides of triangle in any order.
So, putting values from above, we’ll get:
$
\Rightarrow A = \sqrt {15\left( {15 - 12} \right)\left( {15 - 12} \right)\left( {15 - 6} \right)} , \\
\Rightarrow A = \sqrt {15 \times 3 \times 3 \times 9} , \\
\Rightarrow A = 3 \times 3 \times \sqrt {15} , \\
\Rightarrow A = 9\sqrt {15} \\
$
Thus the area of the triangle is $9\sqrt {15} c{m^2}$.
Note:
We can also use $Area = \dfrac{1}{2} \times b \times h$ to find out its area.
For an isosceles triangle, base is always the unequal side. So, in this case $b = 6cm$.
And we can easily find out the height of an isosceles using Pythagoras Theorem.
$
\Rightarrow h = \sqrt {{{12}^2} - {{\left( {\dfrac{b}{2}} \right)}^2}} , \\
\Rightarrow h = \sqrt {144 - 9} , \\
\Rightarrow h = \sqrt {135} , \\
\Rightarrow h = 3\sqrt {15} \\
$
Now, we can use $\dfrac{1}{2} \times b \times h$. We’ll get the same result.
Recently Updated Pages
If a spring has a period T and is cut into the n equal class 11 physics CBSE

A planet moves around the sun in nearly circular orbit class 11 physics CBSE

In any triangle AB2 BC4 CA3 and D is the midpoint of class 11 maths JEE_Main

In a Delta ABC 2asin dfracAB+C2 is equal to IIT Screening class 11 maths JEE_Main

If in aDelta ABCangle A 45circ angle C 60circ then class 11 maths JEE_Main

If in a triangle rmABC side a sqrt 3 + 1rmcm and angle class 11 maths JEE_Main

Trending doubts
Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

A Short Paragraph on our Country India
