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Posted by Pratham Dahiya 6 years, 5 months ago
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Sia ? 6 years, 5 months ago
According to the question, side of the square = 12 cm
Radius of each arc drawn = 6 cm
We know that, Area of quadrant = {tex}\frac{1}{4} \pi r^2{/tex}
Area of one quadrant = {tex}\frac{1}{4} \pi 6^2 = 28.26{/tex}
Area of four quadrants = 4 {tex}\times{/tex} 28.26 = 113.04 cm2
Area of square ABCD = 12 {tex}\times{/tex} 12 = 144 cm2
Hence Area of shaded region {tex}= 144 - 113.04 = 30.96{/tex}cm2
Posted by Shivay Singh Sisodiya 7 years, 9 months ago
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Posted by Praveen Kapoor 6 years, 5 months ago
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Sia ? 6 years, 5 months ago
Let shortest side be x units and other side be y units
Hypotenuse = z units
As per given condition
The perimeter of right-angled triangle is five times the length of its shortest side.
So, x + y + z = 5x
{tex}\Rightarrow{/tex}y + z= 4x
{tex}\Rightarrow{/tex} z = 4x - y ...(i)
The numerical value of the area of the triangle is 15 times the numerical value of the length of the shortest side.
So, area of the rectangle is = 15x
{tex}\Rightarrow{/tex}{tex}\frac{1}{2}{/tex}x{tex}\cdot{/tex}y = 15x
{tex}\Rightarrow{/tex} y = 30 ....(ii)
Using Pythagoras Theorem, we get
z2 = x2 + y2
{tex}\Rightarrow{/tex}(4x - y)2 = x2 + y2 [from (i)]
{tex}\Rightarrow{/tex}(4x - 30)2 = x2 + (30)2 [Using (ii)]
{tex}\Rightarrow{/tex}16x2 - 240x + 900 = x2 + 900
{tex}\Rightarrow{/tex}15x2 - 240x = 0
{tex}\Rightarrow{/tex}15x(x - 16) = 0
{tex}\Rightarrow{/tex}x = 0(rejecting) or x = 16
{tex}\therefore{/tex}x = 16
{tex}\therefore{/tex}length of the shortest side = 16 units
length of other side = 30 units
length of hypotenuse z = 4 {tex}\times{/tex} 16 - 30 = 34 units
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