[tex]\bold{\huge{\underline{ Solution }}}[/tex]
Here, We have given
- 2 squares , In which 1 square is enclosed within the another square and it arranged in a form that it forms 4 right angled triangle
- The height and base of the given right angled triangles are 6 and 3 each.
We know that,
Area of triangle
[tex]{\sf{=}}{\sf{\dfrac{1}{2}}}{\sf{ {\times} base {\times} height }}[/tex]
Subsitute the required values,
[tex]{\sf{=}}{\sf{\dfrac{1}{2}}}{\sf{ {\times} 3 {\times} 6}}[/tex]
[tex]\sf{ = 3 {\times} 3 }[/tex]
[tex]\bold{ = 9 }[/tex]
Therefore,
Area covered by 4 right angled triangles
[tex]\sf{ = 4 {\times} 9 }[/tex]
[tex]\bold{ = 36}[/tex]
Now,
We have to find the area of the big square
- The length of the side of the big square
- [tex]\sf{ = 6 + 3 = 9 }[/tex]
We know that,
Area of square
[tex]\sf{ = Side {\times} Side }[/tex]
Subsitute the required values,
[tex]\sf{ = 9 {\times} 9 }[/tex]
[tex]\bold{ = 81 }[/tex]
Therefore,
The total area of shaded region
= Area of big square - Area covered by 4 right angled triangle
[tex]\sf{ = 81 - 36 }[/tex]
[tex]\bold{ = 45 }[/tex]
Hence, The total area of shaded region is 45 .
Part 2 :-
Here,
We have to find the area of non shaded region
According to the question
- Hypotenuse = The length of square
Let the hypotenuse of the given right angled triangle be x
Therefore,
By using Pythagoras theorem,
- This theorem states that the sum of the squares of the base and perpendicular height is equal to the square of hypotenuse.
That is,
[tex]\sf{ (Perpendicular)^{2} + (Base)^{2} = (Hypotenuse)^{2} }[/tex]
Subsitute the required values
[tex]\sf{ (6)^{2} + (3)^{2} = (x)^{2} }[/tex]
[tex]\sf{ 36 + 9 = (x)^{2} }[/tex]
[tex]\sf{ x = \sqrt{45}}[/tex]
[tex]\bold{ x = 6.7 }[/tex]
That means,
- The length of the small square = 6.7
We know that ,
Area of square
[tex]\sf{ = Side {\times} Side }[/tex]
Subsitute the required values,
[tex]\sf{ = 6.7 {\times} 6.7 }[/tex]
[tex]\bold{ = 44.89 \:\: or \:\: 44.9 }[/tex]
Therefore ,
Area of non shaded region
= Area of big square - Area of small square
[tex]\sf{ = 81 - 44.9 }[/tex]
[tex]\bold{ = 36.1 }[/tex]
Hence, The total area of non shaded region is 36.1 or 36 (approx) .
Part 3 :-
Here, we have to
- find the total area of the figure
Therefore,
The total area of the figure
= Non shaded region + Shaded region
[tex]\sf{= 36 + 45 }[/tex]
[tex]\bold{= 81}[/tex]
Hence, The total area of the given figure is 81 .
