If in a right angled isosceles triangle area of the triangle is 32 sq.cm. find all the sides of the isosceles triangle.
Answer
360.9k+ views
Hint: In this question, the area of the triangle is given and also it is given that the triangle is a right angled isosceles triangle. So the method is that first of all we will find the area of the given triangle by assuming the sides as variable and then equate it to 32. On solving the equation we will get the sides.
Complete step-by-step solution -
In the question, it is given that area of the right angled isosceles triangle is 32 sq.cm.
And we have to find the sides of the triangle.
The diagram for question is shown below:
In the above figure the triangle is a right angled isosceles triangle.
Let us assume that each of two equal sides = x cm.
We know that according to Pythagoras theorem in the $\vartriangle {\text{ABC}}$ , we can write:
${\text{A}}{{\text{B}}^2} + {\text{B}}{{\text{C}}^2} = {\text{A}}{{\text{C}}^2}$ .
Putting the values of AB and BC, we get:
$
{{\text{x}}^2} + {{\text{x}}^2} = {\text{A}}{{\text{C}}^2} \\
\Rightarrow {\text{AC = }}\sqrt {2{{\text{x}}^2}} = \sqrt 2 {\text{x}} \\
$
We know that area of a right triangle is given as:
${\text{Area of }}\vartriangle {\text{ABC = }}\dfrac{1}{2} \times {\text{base}} \times {\text{height = }}\dfrac{1}{2} \times {\text{BC}} \times {\text{AB}}$ .
Putting the values of AB and BC in above equation, We get:
${\text{Area of }}\vartriangle {\text{ABC = }}\dfrac{1}{2} \times {\text{BC}} \times {\text{AB = }}\dfrac{1}{2} \times {\text{x}} \times {\text{x = }}\dfrac{{{{\text{x}}^2}}}{2}{\text{c}}{{\text{m}}^2}$
Now according to the question, we can write:
$
\dfrac{{{{\text{x}}^2}}}{2} = 32 \\
\Rightarrow {{\text{x}}^2} = 64 \\
\Rightarrow {\text{x = }}\sqrt {64} = 8 \\
$
Therefore, the length of the side AB=8cm.
And the length of the side BC= 8cm.
Length of side AC = $\sqrt 2 \times {\text{x = }}\sqrt 2 \times 8$ cm =11.31cm.
Note: In this type of question where the area of a triangle is given and the sides length is asked. First step is to draw the diagram of the question then assume the unknown side length. In this question the triangle is a right angled isosceles triangle. So you should know the area of the isosceles right triangle.
Complete step-by-step solution -
In the question, it is given that area of the right angled isosceles triangle is 32 sq.cm.
And we have to find the sides of the triangle.
The diagram for question is shown below:

In the above figure the triangle is a right angled isosceles triangle.
Let us assume that each of two equal sides = x cm.
We know that according to Pythagoras theorem in the $\vartriangle {\text{ABC}}$ , we can write:
${\text{A}}{{\text{B}}^2} + {\text{B}}{{\text{C}}^2} = {\text{A}}{{\text{C}}^2}$ .
Putting the values of AB and BC, we get:
$
{{\text{x}}^2} + {{\text{x}}^2} = {\text{A}}{{\text{C}}^2} \\
\Rightarrow {\text{AC = }}\sqrt {2{{\text{x}}^2}} = \sqrt 2 {\text{x}} \\
$
We know that area of a right triangle is given as:
${\text{Area of }}\vartriangle {\text{ABC = }}\dfrac{1}{2} \times {\text{base}} \times {\text{height = }}\dfrac{1}{2} \times {\text{BC}} \times {\text{AB}}$ .
Putting the values of AB and BC in above equation, We get:
${\text{Area of }}\vartriangle {\text{ABC = }}\dfrac{1}{2} \times {\text{BC}} \times {\text{AB = }}\dfrac{1}{2} \times {\text{x}} \times {\text{x = }}\dfrac{{{{\text{x}}^2}}}{2}{\text{c}}{{\text{m}}^2}$
Now according to the question, we can write:
$
\dfrac{{{{\text{x}}^2}}}{2} = 32 \\
\Rightarrow {{\text{x}}^2} = 64 \\
\Rightarrow {\text{x = }}\sqrt {64} = 8 \\
$
Therefore, the length of the side AB=8cm.
And the length of the side BC= 8cm.
Length of side AC = $\sqrt 2 \times {\text{x = }}\sqrt 2 \times 8$ cm =11.31cm.
Note: In this type of question where the area of a triangle is given and the sides length is asked. First step is to draw the diagram of the question then assume the unknown side length. In this question the triangle is a right angled isosceles triangle. So you should know the area of the isosceles right triangle.
Last updated date: 28th Sep 2023
•
Total views: 360.9k
•
Views today: 11.60k
Recently Updated Pages
What do you mean by public facilities

Slogan on Noise Pollution

Paragraph on Friendship

Disadvantages of Advertising

Prepare a Pocket Guide on First Aid for your School

What is the Full Form of ILO, UNICEF and UNESCO

Trending doubts
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Summary of the poem Where the Mind is Without Fear class 8 english CBSE

Difference Between Plant Cell and Animal Cell

What is the basic unit of classification class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

One cusec is equal to how many liters class 8 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Give 10 examples for herbs , shrubs , climbers , creepers
