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Question
The hypotenuse of a right triangle is `3sqrt10`. If the smaller leg is tripled and the longer leg doubled, new hypotenuse wll be `9sqrt5`. How long are the legs of the triangle?
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Solution
Let the length of smaller side of right triangle be x cm then larger side be y cm
Then, as we know that by Pythagoras theorem
`x^2 + y^2 = (3sqrt10)^2`
x2 + y2 = 90 .............. (1)
If the smaller side is triple and the larger side be doubled, the new hypotenuse is `9sqrt5` cm
Therefore,
`(3x)^2+(2y)^2=(9sqrt5)^2`
9x2 + 4y2 = 405 ............. (2)
From equation (1) we get y2 = 90 - x2
Now putting the value of y2 in equation (2)
9x2 + 4(90 - x2) = 405
9x2 + 360 - 4x2 - 405 = 0
5x2 - 45 = 0
5(x2 - 9) = 0
x2 - 9 = 0
x2 = 9
`x = sqrt9` = ± 3
But, the side of right triangle can never be negative
Therefore, when x = 3 then
y2 = 90 - x2
= 90 - (3)2
= 90 - 9
= 81
`y=sqrt81`
= ± 9
Hence, length of smaller side of right triangle be 3 cm then larger side be 9 cm
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