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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? - CBSE Class 10 - Mathematics

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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?

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

  Is there an error in this question or solution?

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Solution 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? Concept: Solutions of Quadratic Equations by Factorization.
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