Question

area of right triangle is 24 sq cm and the square if sum of the perpendicular sides is 196 taking perpendicular sides as x and y metres then find what is x+y and find the number x-y and find length of sides of right triangle​

Answer 1

Area of the triangle =24 centimeter square

square of the sum of perpendicular sides =196

(x+y) square=196

x+y= root 196

x+y=14

1 ×x×y=24

2

1 xy =24

2

xy=2×24

xy=48

(x-y)square =(x+y)square-4xy

=196 -4×24

=196-192

(x-y)square = root 4

x-y=2

x=14+2 =16 =8

2 2

y=14-2 =12= 6

2 2

Answer 2

Given,

Area of right angle triangle $\[ = 24c{m^2}\]$

$\[{x^2} + {y^2} = 196\]$---------(1)

Solution,

$\[{x^2} + {y^2} = 196\]$---------(1)

$\[\begin{array}{l}\frac{1}{2}xy = 24\\ \Rightarrow xy = 48\end{array}\]$---------(2)

Calculate the value of $\[\left( {x + y} \right)\]$.

$\[\begin{array}{l}\left( {x + y} \right) = \sqrt {{{\left( {x + y} \right)}^2}} \\ \Rightarrow \left( {x + y} \right) = \sqrt {{x^2} + {y^2} + 2xy} \\ \Rightarrow \left( {x + y} \right) = \sqrt {196 + 2 \times 48} \\ \Rightarrow \left( {x + y} \right) = \sqrt {196 + 96} \\ \Rightarrow \left( {x + y} \right) = \sqrt {292} \\ \Rightarrow \left( {x + y} \right) = 2\sqrt {73} \end{array}\]$

Calculate the value of $\[\left( {x - y} \right)\]$.

$\[\begin{array}{l}\left( {x - y} \right) = \sqrt {{{\left( {x - y} \right)}^2}} \\ \Rightarrow \left( {x - y} \right) = \sqrt {{x^2} + {y^2} - 2xy} \\ \Rightarrow \left( {x - y} \right) = \sqrt {196 - 2 \times 48} \\ \Rightarrow \left( {x - y} \right) = \sqrt {196 - 96} \\ \Rightarrow \left( {x - y} \right) = \sqrt {100} \\ \Rightarrow \left( {x - y} \right) = 10\end{array}\]$

Calculate the length of sides of triangle.

$\[\begin{array}{l}x + y = 2\sqrt {73} - - - - \left( 3 \right)\\x - y = 10 - - - - - \left( 4 \right)\end{array}\]$

Add equation 3 and 4.

$\[\begin{array}{l}2x = 2\sqrt {73} + 10\\ \Rightarrow x = \sqrt {73} + 5\end{array}\]$

Subtract equation 3 and 4.

$\[\begin{array}{l}2y = 2\sqrt {73} - 10\\ \Rightarrow y = \sqrt {73} - 5\end{array}\]$